Setting up an alert to monitor where/when your databases are backed up
You can use a BackupRestore Event Notification to monitor backup events and this solution even works on Sql Express (where the Sql Agent doesn't exist). This link shows you how to set up SQL Mail for SQL Server Express.

I created a database called Test.

I modified this example from Jonathan Kehayias, which was tracking autogrowth, to instead audit the backup/restore event.

Here is my example that was tweaked from Jonathan's:

-- Using msdb prevents the need for certificate signing the 
-- activation procedure to execute sp_send_dbmail across
-- databases
USE [msdb];
GO

-- Drop the notification if it exists
IF EXISTS (
        SELECT *
        FROM sys.server_event_notifications
        WHERE NAME = N'CaptureDatabaseBackupEvent'
        )
BEGIN
    DROP EVENT NOTIFICATION CaptureDatabaseBackupEvent ON SERVER;
END

-- Drop the route if it exists
IF EXISTS (
        SELECT *
        FROM sys.routes
        WHERE NAME = N'DatabaseBackupEventRoute'
        )
BEGIN
    DROP ROUTE DatabaseBackupEventRoute;
END

-- Drop the service if it exists
IF EXISTS (
        SELECT *
        FROM sys.services
        WHERE NAME = N'DatabaseBackupEventService'
        )
BEGIN
    DROP SERVICE DatabaseBackupEventService;
END

-- Drop the queue if it exists
IF EXISTS (
        SELECT *
        FROM sys.service_queues
        WHERE NAME = N'DatabaseBackupEventQueue'
        )
BEGIN
    DROP QUEUE DatabaseBackupEventQueue;
END

--  Create a service broker queue to hold the events
CREATE QUEUE [DatabaseBackupEventQueue]
    WITH STATUS = ON;
GO

--  Create a service broker service receive the events
CREATE SERVICE [DatabaseBackupEventService] ON QUEUE [DatabaseBackupEventQueue] ([http://schemas.microsoft.com/SQL/Notifications/PostEventNotification]);
GO

-- Create a service broker route to the service
CREATE ROUTE [DatabaseBackupEventRoute]
    WITH SERVICE_NAME = 'DatabaseBackupEventService'
        ,ADDRESS = 'LOCAL';
GO

-- Create the event notification to capture the events
CREATE EVENT NOTIFICATION [CaptureDatabaseBackupEvent] ON SERVER
    WITH FAN_IN
FOR AUDIT_BACKUP_RESTORE_EVENT TO SERVICE 'DatabaseBackupEventService'
    ,'current database';
GO

--  Alter the queue to use the activation procedure
ALTER QUEUE [DatabaseBackupEventQueue]
   WITH STATUS=ON, 
      ACTIVATION 
         (STATUS=ON,
          PROCEDURE_NAME = [ProcessDatabaseBackupEvents],
          MAX_QUEUE_READERS = 1,
          EXECUTE AS OWNER);
GO


I then tweaked his example stored procedure that consumes the event.

Make sure you change this line in the stored procedure to be your email address

@recipients = 'YourEmailAddress'

DROP PROCEDURE [dbo].[ProcessDatabaseBackupEvents];
GO

CREATE PROCEDURE [dbo].[ProcessDatabaseBackupEvents]
    WITH EXECUTE AS OWNER
AS
DECLARE @message_body XML;
DECLARE @message_sequence_number INT;
DECLARE @dialog UNIQUEIDENTIFIER;
DECLARE @email_message NVARCHAR(MAX);

WHILE (1 = 1)
BEGIN
    BEGIN TRANSACTION;

    -- Receive the next available message FROM the queue
    WAITFOR (
            RECEIVE TOP (1) -- just handle one message at a time
            @message_body = CAST(message_body AS XML) FROM dbo.DatabaseBackupEventQueue
            )
        ,TIMEOUT 1000;-- if queue empty for 1 sec, give UPDATE AND GO away

    -- If we didn't get anything, bail out
    IF (@@ROWCOUNT = 0)
    BEGIN
        ROLLBACK TRANSACTION;

        BREAK;
    END

    DECLARE @EventType VARCHAR(128);
    DECLARE @ServerName VARCHAR(128);
    DECLARE @PostTime VARCHAR(128);
    DECLARE @DatabaseName VARCHAR(128);
    DECLARE @TextData VARCHAR(max);
    DECLARE @GrowthPages INT;

    SELECT @EventType = @message_body.value('(/EVENT_INSTANCE/EventType)[1]', 'varchar(128)')
        ,@TextData = @message_body.value('(/EVENT_INSTANCE/TextData)[1]', 'varchar(128)')
        ,@ServerName = @message_body.value('(/EVENT_INSTANCE/ServerName)[1]', 'varchar(128)')
        ,@PostTime = CAST(@message_body.value('(/EVENT_INSTANCE/PostTime)[1]', 'datetime') AS VARCHAR)
        ,@DatabaseName = @message_body.value('(/EVENT_INSTANCE/DatabaseName)[1]', 'varchar(128)');

    -- Generate formatted email message
    SELECT @email_message = 'The following DatabaseBackup event                                      occurred:' + CHAR(10) + CAST('ServerName: ' AS CHAR(25)) + @ServerName + CHAR(10) + CAST('PostTime: ' AS CHAR(25)) + @PostTime + CHAR(10) + CAST('DatabaseName: ' AS CHAR(25)) + @DatabaseName + @TextData

    -- Send email using Database Mail
    EXEC msdb.dbo.sp_send_dbmail @profile_name = 'SqlServerEmailProfile'
        ,
        -- your defined email profile 
        @recipients = 'YourEmailAddress'
        ,-- your email
        @subject = 'DatabaseBackup Event Notification'
        ,@body = @email_message;

    --  Commit the transaction.  At any point before this, we could roll 
    --  back. The received message would be back on the queue AND the 
    --  response wouldn't be sent.
    COMMIT TRANSACTION;
END
GO


Then, I ran a simple backup

BACKUP DATABASE [Test] TO DISK = N'C:\Program Files\Microsoft SQL Server\MSSQL13.MSSQLSERVER\MSSQL\Backup\test.bak'
WITH NOFORMAT
    ,INIT
    ,NAME = N'Test-Full Database Backup'
    ,SKIP
    ,NOREWIND
    ,NOUNLOAD
    ,STATS = 10
GO


After a minute or so, I received the email with the event data in it. You can modify the event consuming stored procedure to capture whatever you need and send that information along in the email.