SELECT *
FROM dbo.tablename
WHERE Field1 >= '20' + field3 + CASE field2 
    WHEN '1Q' THEN '01' 
    WHEN '2Q' THEN '04' 
    WHEN '3Q' THEN '07'
    WHEN '4Q' THEN '10' ELSE '01' END + '01' AND
    Field1 < DATEADD(QUARTER, 1, '20' + field3 + CASE field2 
    WHEN '1Q' THEN '01' 
    WHEN '2Q' THEN '04' 
    WHEN '3Q' THEN '07'
    WHEN '4Q' THEN '10' ELSE '01' END + '01')

IF object_id('tempdb..#tbl') IS NOT NULL
BEGIN
   DROP TABLE #tbl
END

-- Assumptions
-- 1) Field2 <= Field3
-- 2) Valid ranges for Field2 and Field3 are in the expected format and between [2000, 2099]
-- 3) Field1 values can be converted to valid dates between [1/1/2000, 12/31/2099]
-- 4) You are unable to implement a more appropriate design to the table at this time

-- Create some test data
CREATE TABLE #tbl (
	Field1 VARCHAR(8),
	Field2 VARCHAR(4),
	Field3 VARCHAR(4)
)

INSERT INTO #tbl(Field1, Field2, Field3) 
VALUES('20120908', '4Q12', '1Q13')  -- Field1 is before range
	, ('20121108', '4Q12', '1Q13')	-- Field1 is in range
	, ('20130408', '4Q12', '1Q13')	-- Field1 is after range

-- Test select	
SELECT *
FROM #tbl
WHERE RIGHT(LEFT(Field1, 4), 2) + 'Q' + CAST((MONTH(CONVERT(DATE, Field1, 112)) - 1) / 3 + 1 AS VARCHAR)
	BETWEEN RIGHT(Field2, 2) + 'Q' + LEFT(Field2, 1) AND RIGHT(Field3, 2) + 'Q' + LEFT(Field3, 1)