Get Quarterly Data For a Year
Using your CASE expression example, here's how it could be written:

DECLARE @PaymentHistory TABLE (
    EmployeeName VARCHAR(100)
    ,DatePaid DATE
    ,TotalCheckAmt DECIMAL(11, 2)
    )
insert into @Paymenthistory values ('James','2016-01-01',100.00)
insert into @Paymenthistory values ('James','2016-02-01',100.00)
insert into @Paymenthistory values ('James','2016-03-01',100.00)
insert into @Paymenthistory values ('James','2016-04-01',100.00)
insert into @Paymenthistory values ('James','2016-05-01',100.00)
insert into @Paymenthistory values ('James','2016-06-01',100.00)
insert into @Paymenthistory values ('James','2016-07-01',100.00)

insert into @Paymenthistory values ('Roger','2016-01-01',100.00)
insert into @Paymenthistory values ('Roger','2016-02-01',100.00)
insert into @Paymenthistory values ('Roger','2016-03-01',100.00)
insert into @Paymenthistory values ('Roger','2016-04-01',100.00)
insert into @Paymenthistory values ('Roger','2016-05-01',100.00)
insert into @Paymenthistory values ('Roger','2016-06-01',100.00)
insert into @Paymenthistory values ('Roger','2016-07-01',100.00)
SELECT Employeename
    ,ISNULL(SUM(CASE 
                WHEN Datepart(Year, [DatePaid]) = '2016'
                    AND DATEPART(Quarter, [DatePaid]) = 1
                    THEN TotalCheckamt
                END), 0) AS Q12016
    ,ISNULL(SUM(CASE 
                WHEN Datepart(Year, [DatePaid]) = '2017'
                    AND DATEPART(Quarter, [DatePaid]) = 1
                    THEN TotalCheckamt
                END), 0) AS Q12017
    ,ISNULL(SUM(CASE 
                WHEN Datepart(Year, [DatePaid]) = '2016'
                    AND DATEPART(Quarter, [DatePaid]) = 2
                    THEN TotalCheckamt
                END), 0) AS Q22016
    ,ISNULL(SUM(CASE 
                WHEN Datepart(Year, [DatePaid]) = '2017'
                    AND DATEPART(Quarter, [DatePaid]) = 2
                    THEN TotalCheckamt
                END), 0) AS Q22017
    ,ISNULL(SUM(CASE 
                WHEN Datepart(Year, [DatePaid]) = '2016'
                    AND DATEPART(Quarter, [DatePaid]) = 3
                    THEN TotalCheckamt
                END), 0) AS Q32016
    ,ISNULL(SUM(CASE 
                WHEN Datepart(Year, [DatePaid]) = '2017'
                    AND DATEPART(Quarter, [DatePaid]) = 3
                    THEN TotalCheckamt
                END), 0) AS Q32017
    ,ISNULL(SUM(CASE 
                WHEN Datepart(Year, [DatePaid]) = '2016'
                    AND DATEPART(Quarter, [DatePaid]) = 4
                    THEN TotalCheckamt
                END), 0) AS Q42016
    ,ISNULL(SUM(CASE 
                WHEN Datepart(Year, [DatePaid]) = '2017'
                    AND DATEPART(Quarter, [DatePaid]) = 4
                    THEN TotalCheckamt
                END), 0) AS Q42017
FROM @PaymentHistory
GROUP BY Employeename
ORDER BY Employeename ASC


After you accepted my original answer, I noticed where other posters were advocating the use of PIVOT to achieve the same result as multiple CASE expressions.  For completeness of my answer, I decided to try my hand at using PIVOT to provide additional options for you (and learn something myself in the process).  I've found the information from Questions About Pivoting Data in SQL Server You Were Too Shy to Ask to be quite useful when trying to understand how to use PIVOT.  You'll learn a lot about PIVOT by working through the examples (static and dynamic PIVOT) in the link - I know I did. 

To summarize some of the key points of PIVOT (from the link):


  You use the PIVOT operator within your query’s FROM clause to rotate
  and aggregate the values in a dataset. The data is pivoted based on
  one of the columns in the dataset. Each unique value in that column
  becomes its own column, which contains aggregated pivoted data.
  
  To better understand how this all works, let’s start with the basic
  syntax for a query that uses the PIVOT operator:


SELECT column_list
FROM table_expression
  PIVOT
  (
    aggregate_function(aggregate_column)
    FOR pivot_column
    IN( pivot_column_values )
  ) [AS] pivot_table_alias
[ORDER BY column_list];



  For the SELECT clause, you can specify an asterisk (*) or the
  individual columns, and for the FROM clause, you can specify a table
  or table expression. If you use a table expression, then you must also
  define a table alias. You can also include an ORDER BY clause, but
  that’s optional. You’ll get to see these clauses in action as we
  progress through the questions. For now, let’s focus on the PIVOT
  clause. You need to understand how this clause works to make sure your
  pivots work the way you want.
  
  After you specify the PIVOT keyword, you pass in what are essentially
  three arguments, enclosed in parentheses. The first is the aggregate
  function and the name of the column to be aggregated. You can use any
  aggregate function except the COUNT function, when used with an
  asterisk, as in COUNT(*).
  
  Next, you define the FOR subclause, which specifies the column on
  which the pivot will be based. It is this column whose distinct values
  are turned into their own columns. The FOR subclause also includes the
  IN operator, which is where you specify the pivot column values that
  will be turned into columns. The values you specify here must exist in
  the pivot column or they will be ignored.


I'm by no means a PIVOT expert and other posters may have better solutions (hey, I'm willing to learn a better way), but here's my take on it.

I'm including two examples of PIVOT - one static and one dynamic.
(Hopefully, by using my examples and the information in the link provided, you'll understand what I'm doing)

The first example is a static PIVOT and is similar to the CASE expression solution I originally provided (except it uses PIVOT).

Notice that my sample data now includes years 'outside' of the CASE expression.  That information won't show up in the static version, but you'll see how we can solve that in the dynamic version later.

Static PIVOT

DECLARE @PaymentHistory TABLE (
    EmployeeName VARCHAR(100)
    ,DatePaid DATE
    ,TotalCheckAmt DECIMAL(11, 2)
    );
insert into @Paymenthistory values ('James','2016-01-01',100.00);
insert into @Paymenthistory values ('James','2016-02-01',100.00);
insert into @Paymenthistory values ('James','2016-03-01',100.00);
insert into @Paymenthistory values ('James','2016-04-01',100.00);
insert into @Paymenthistory values ('James','2016-05-01',100.00);
insert into @Paymenthistory values ('James','2016-06-01',100.00);
insert into @Paymenthistory values ('James','2016-07-01',100.00);
insert into @Paymenthistory values ('James','2017-01-01',100.00);
insert into @Paymenthistory values ('James','2018-10-01',900.00);

insert into @Paymenthistory values ('Roger','2016-01-01',100.00);
insert into @Paymenthistory values ('Roger','2016-02-01',100.00);
insert into @Paymenthistory values ('Roger','2016-03-01',100.00);
insert into @Paymenthistory values ('Roger','2016-04-01',100.00);
insert into @Paymenthistory values ('Roger','2016-05-01',100.00);
insert into @Paymenthistory values ('Roger','2016-06-01',100.00);
insert into @Paymenthistory values ('Roger','2016-07-01',100.00);
insert into @Paymenthistory values ('Roger','2020-10-01',900.00);
;
WITH cte_Paymenthistory
AS (
    SELECT CASE 
            WHEN Datepart(Year, [DatePaid]) = '2016'
                AND DATEPART(Quarter, [DatePaid]) = 1
                THEN 'Q12016'
            WHEN Datepart(Year, [DatePaid]) = '2016'
                AND DATEPART(Quarter, [DatePaid]) = 2
                THEN 'Q22016'
            WHEN Datepart(Year, [DatePaid]) = '2016'
                AND DATEPART(Quarter, [DatePaid]) = 3
                THEN 'Q32016'
            WHEN Datepart(Year, [DatePaid]) = '2016'
                AND DATEPART(Quarter, [DatePaid]) = 4
                THEN 'Q42016'
            WHEN Datepart(Year, [DatePaid]) = '2017'
                AND DATEPART(Quarter, [DatePaid]) = 2
                THEN 'Q12017'
            WHEN Datepart(Year, [DatePaid]) = '2017'
                AND DATEPART(Quarter, [DatePaid]) = 3
                THEN 'Q32017'
            WHEN Datepart(Year, [DatePaid]) = '2017'
                AND DATEPART(Quarter, [DatePaid]) = 4
                THEN 'Q42017'
            END AS ColumnLabel
        ,EmployeeName
        ,TotalCheckAmt
    FROM @PaymentHistory
    )
SELECT EmployeeName
    ,coalesce([Q12016], 0) AS [Q12016]
    ,coalesce([Q12017], 0) AS [Q12076]
    ,coalesce([Q22016], 0) AS [Q22016]
    ,coalesce([Q22017], 0) AS [Q12017]
    ,coalesce([Q32016], 0) AS [Q32016]
    ,coalesce([Q32017], 0) AS [Q32017]
    ,coalesce([Q42016], 0) AS [Q42016]
    ,coalesce([Q42017], 0) AS [Q42017]
FROM cte_Paymenthistory
PIVOT(SUM(TotalCheckAmt) FOR ColumnLabel IN (
            [Q12016]
            ,[Q12017]
            ,[Q22016]
            ,[Q22017]
            ,[Q32016]
            ,[Q32017]
            ,[Q42016]
            ,[Q42017]
            )) AS pvt
ORDER BY employeename;


Did you notice the data outside the CASE expressions didn't show up?  

Let's try to solve that by using a dynamic PIVOT.

Dynamic PIVOT


  How do I pivot data if I don’t know the values in the pivoted column?


(Due to using dynamic SQL, we have to switch to using a temporary table instead of a table variable)

set nocount on
DECLARE @sql AS NVARCHAR(2000);
DECLARE @col AS NVARCHAR(2000);
DECLARE @colCoalesceNull AS NVARCHAR(2000);

IF OBJECT_ID('tempdb..#PaymentHistory') IS NOT NULL drop Table #PaymentHistory
CREATE TABLE #PaymentHistory (
    EmployeeName VARCHAR(100)
    ,DatePaid DATE
    ,TotalCheckAmt DECIMAL(11, 2)
    );
insert into #Paymenthistory values ('James','2016-01-01',100.00);
insert into #Paymenthistory values ('James','2016-02-01',100.00);
insert into #Paymenthistory values ('James','2016-03-01',100.00);
insert into #Paymenthistory values ('James','2016-04-01',100.00);
insert into #Paymenthistory values ('James','2016-05-01',100.00);
insert into #Paymenthistory values ('James','2016-06-01',100.00);
insert into #Paymenthistory values ('James','2016-07-01',100.00);
insert into #Paymenthistory values ('James','2017-01-01',100.00);
insert into #Paymenthistory values ('James','2018-10-01',900.00);

insert into #Paymenthistory values ('Roger','2016-01-01',100.00);
insert into #Paymenthistory values ('Roger','2016-02-01',100.00);
insert into #Paymenthistory values ('Roger','2016-03-01',100.00);
insert into #Paymenthistory values ('Roger','2016-04-01',100.00);
insert into #Paymenthistory values ('Roger','2016-05-01',100.00);
insert into #Paymenthistory values ('Roger','2016-06-01',100.00);
insert into #Paymenthistory values ('Roger','2016-07-01',100.00);
insert into #Paymenthistory values ('Roger','2020-10-01',900.00);
;
;
SELECT @col = 
        Coalesce(@col + ', ', '') + QUOTENAME(PvtColumnName)
    ,@colCoalesceNull = 
        Coalesce(@colCoalesceNull + ', ', '') + 'coalesce(' + QUOTENAME(PvtColumnName) + ',0) as ' + QUOTENAME(PvtColumnName)
FROM (
    SELECT DISTINCT 
    'Q' + 
    CONVERT(VARCHAR(1), DATEPART(Quarter, [DatePaid])) + 
    CONVERT(VARCHAR(4), year([DatePaid])) 
        AS PvtColumnName
    FROM #PaymentHistory
    ) AS PaymentHistory;

PRINT @col
PRINT @colcoalescenull

SET @sql = N'
with cte_PaymentHistory as
(
select 
''Q'' + CONVERT(varchar(1),DATEPART(Quarter, [DatePaid])) + CONVERT(varchar(4),year([DatePaid])) as PvtColumnName
,EmployeeName
,TotalCheckAmt
from #PaymentHistory
)
SELECT EmployeeName, ' + @colCoalesceNull + 'FROM cte_PaymentHistory
    PIVOT(SUM(TotalCheckAmt)
    FOR PvtColumnName IN (' + @col + ')) AS PivotPaymentHistory';

EXEC sp_executesql @sql;


Your question prompted me to research and learn more about PIVOT - thanks.