Do stored procedures prevent SQL injection? Is it true that stored procedures prevent SQL injection attacks against PostgreSQL databases? I did a little research and found out that SQL Server, Oracle and MySQL are not safe against SQL injection even if we only use stored procedures. However, this problem does not exist in PostgreSQL. 

Does the stored procedure implementation in PostgreSQL core prevent SQL injection attacks or is it something else? Or is PostgreSQL also susceptible to SQL injection even if we only use stored procedures? If so, please show me an example (e.g. book, site, paper, etc).
No, stored procedures do not prevent SQL injection.  Here's an actual example (from an in-house app someone created where I work) of a stored procedure that unfortunately permits SQL injection:

This sql server code:

CREATE PROCEDURE [dbo].[sp_colunmName2]   
    @columnName as nvarchar(30),
    @type as nvarchar(30), 
    @searchText as nvarchar(30)           
AS
BEGIN
    DECLARE @SQLStatement NVARCHAR(4000)
    BEGIN
        SELECT @SQLStatement = 'select * from Stations where ' 
            + @columnName + ' ' + @type + ' ' + '''' + @searchText + '''' 
        EXEC(@SQLStatement)
    END      
END
GO


roughly equivalent to postgres:

CREATE or replace FUNCTION public.sp_colunmName2 (
    columnName  varchar(30),
    type varchar(30), 
    searchText  varchar(30) ) RETURNS SETOF stations LANGUAGE plpgsql            
AS
$$
DECLARE SQLStatement VARCHAR(4000);
BEGIN
    SQLStatement = 'select * from Stations where ' 
            || columnName || ' ' || type || ' ' || ''''|| searchText || '''';
    RETURN QUERY EXECUTE  SQLStatement;
END
$$;


The developer's idea was to create a versatile search procedure, but the result is that the WHERE clause can contain anything the user wants, allowing a visit from little Bobby Tables.

Whether you use SQL statements or stored procedure doesn't matter.  What matters is whether your SQL uses parameters or concatenated strings.  Parameters prevent SQL injection; concatenated strings allow SQL injection.