declare @table table
(
[Paid Date] date
)
insert into @table
values('20150102'),('20150512'),('20150830'),('20151231'),('20141230')
;WITH Quarters AS (
   SELECT Q = 'Q1', MonthBegin = 1, MonthEnd = 3 UNION
   SELECT Q = 'Q2', MonthBegin = 4, MonthEnd = 6 UNION
   SELECT Q = 'Q3', MonthBegin = 7, MonthEnd = 9 UNION
   SELECT Q = 'Q4', MonthBegin = 10, MonthEnd = 12
)
SELECT
   [paid date],[Quarter] = 'FSY'+CONVERT(VARCHAR(4), DATEPART(yyyy, CONVERT(DATETIME, [paid date]))) + '_' + q.Q
FROM
@table
  INNER JOIN Quarters q ON
      DATEPART(m, CONVERT(DATETIME, [paid date])) >= q.MonthBegin AND
      DATEPART(m, CONVERT(DATETIME, [paid date])) <= q.MonthEnd;

Output