Two functions that return the date of the start and end of the quarter of which the given date is in. (+some "unit tests").

qStartEnd.sql

DELIMITER $$
/** qStart (DATE date)
 *  Get the date of the start of a quarter given a date in that quarter.
 */

DROP FUNCTION IF EXISTS `qStart`$$
CREATE FUNCTION `qStart`(d DATE) RETURNS DATE
    DETERMINISTIC
BEGIN
  RETURN d - INTERVAL MOD(MONTH(d)-1,3) MONTH - INTERVAL DAY(d)-1 DAY;
END$$

/** qEnd (DATE date)
 *  Get the date of the end of a quarter given a date in that quarter.
 *  Please note that this could be simplified if you use have the qStart function 
 */
DROP FUNCTION IF EXISTS `qEnd`$$
CREATE FUNCTION `qEnd`(d DATE) RETURNS DATE
    DETERMINISTIC
BEGIN
	DECLARE quarterDateStart DATE DEFAULT d+INTERVAL 3 MONTH;
	RETURN (quarterDateStart - INTERVAL MOD(MONTH(quarterDateStart)-1,3) MONTH - INTERVAL DAY(quarterDateStart)-1 DAY) - INTERVAL 1 DAY;
	/**  Please note that this could be simplified if you use have the qStart function like so:
	 * RETURN qStart(d+INTERVAL 3 MONTH) - INTERVAL 1 DAY
	 */
END$$
DELIMITER ;

SELECT 

IF(qStart(DATE('2013-01-01')) = DATE('2013-01-01'),"OK",CONCAT(qStart(DATE('2013-01-01'))," != ",DATE('2013-01-01'))) as qs_a,
IF(qStart(DATE('2013-02-01')) = DATE('2013-01-01'),"OK",CONCAT(qStart(DATE('2013-02-01'))," != ",DATE('2013-01-01'))) as qs_b,
IF(qStart(DATE('2013-03-01')) = DATE('2013-01-01'),"OK",CONCAT(qStart(DATE('2013-03-01'))," != ",DATE('2013-01-01'))) as qs_c,
IF(qStart(DATE('2013-04-01')) = DATE('2013-04-01'),"OK",CONCAT(qStart(DATE('2013-04-01'))," != ",DATE('2013-04-01'))) as qs_d,
IF(qStart(DATE('2013-05-01')) = DATE('2013-04-01'),"OK",CONCAT(qStart(DATE('2013-05-01'))," != ",DATE('2013-04-01'))) as qs_e,
IF(qStart(DATE('2013-06-01')) = DATE('2013-04-01'),"OK",CONCAT(qStart(DATE('2013-06-01'))," != ",DATE('2013-04-01'))) as qs_f,
IF(qStart(DATE('2013-07-01')) = DATE('2013-07-01'),"OK",CONCAT(qStart(DATE('2013-07-01'))," != ",DATE('2013-07-01'))) as qs_g,
IF(qStart(DATE('2013-08-01')) = DATE('2013-07-01'),"OK",CONCAT(qStart(DATE('2013-08-01'))," != ",DATE('2013-07-01'))) as qs_h,
IF(qStart(DATE('2013-09-01')) = DATE('2013-07-01'),"OK",CONCAT(qStart(DATE('2013-09-01'))," != ",DATE('2013-07-01'))) as qs_i,
IF(qStart(DATE('2013-10-01')) = DATE('2013-10-01'),"OK",CONCAT(qStart(DATE('2013-10-01'))," != ",DATE('2013-10-01'))) as qs_j,
IF(qStart(DATE('2013-11-01')) = DATE('2013-10-01'),"OK",CONCAT(qStart(DATE('2013-11-01'))," != ",DATE('2013-10-01'))) as qs_k,
IF(qStart(DATE('2013-12-01')) = DATE('2013-10-01'),"OK",CONCAT(qStart(DATE('2013-12-01'))," != ",DATE('2013-10-01'))) as qs_l,

IF(qStart(DATE('2013-01-18')) = DATE('2013-01-01'),"OK",CONCAT(qStart(DATE('2013-01-18'))," != ",DATE('2013-01-01'))) as qs_ab,
IF(qStart(DATE('2013-02-18')) = DATE('2013-01-01'),"OK",CONCAT(qStart(DATE('2013-02-18'))," != ",DATE('2013-01-01'))) as qs_bb,
IF(qStart(DATE('2013-03-18')) = DATE('2013-01-01'),"OK",CONCAT(qStart(DATE('2013-03-18'))," != ",DATE('2013-01-01'))) as qs_cb,
IF(qStart(DATE('2013-04-18')) = DATE('2013-04-01'),"OK",CONCAT(qStart(DATE('2013-04-18'))," != ",DATE('2013-04-01'))) as qs_db,
IF(qStart(DATE('2013-05-18')) = DATE('2013-04-01'),"OK",CONCAT(qStart(DATE('2013-05-18'))," != ",DATE('2013-04-01'))) as qs_eb,
IF(qStart(DATE('2013-06-18')) = DATE('2013-04-01'),"OK",CONCAT(qStart(DATE('2013-06-18'))," != ",DATE('2013-04-01'))) as qs_fb,
IF(qStart(DATE('2013-07-18')) = DATE('2013-07-01'),"OK",CONCAT(qStart(DATE('2013-07-18'))," != ",DATE('2013-07-01'))) as qs_gb,
IF(qStart(DATE('2013-08-18')) = DATE('2013-07-01'),"OK",CONCAT(qStart(DATE('2013-08-18'))," != ",DATE('2013-07-01'))) as qs_hb,
IF(qStart(DATE('2013-09-18')) = DATE('2013-07-01'),"OK",CONCAT(qStart(DATE('2013-09-18'))," != ",DATE('2013-07-01'))) as qs_ib,
IF(qStart(DATE('2013-10-18')) = DATE('2013-10-01'),"OK",CONCAT(qStart(DATE('2013-10-18'))," != ",DATE('2013-10-01'))) as qs_jb,
IF(qStart(DATE('2013-11-18')) = DATE('2013-10-01'),"OK",CONCAT(qStart(DATE('2013-11-18'))," != ",DATE('2013-10-01'))) as qs_kb,
IF(qStart(DATE('2013-12-18')) = DATE('2013-10-01'),"OK",CONCAT(qStart(DATE('2013-12-18'))," != ",DATE('2013-10-01'))) as qs_lb,

IF(qEnd(DATE('2013-01-01')) = DATE('2013-03-31'),"OK",CONCAT(qEnd(DATE('2013-01-01'))," != ",DATE('2013-03-31'))) as qe_a,
IF(qEnd(DATE('2013-02-01')) = DATE('2013-03-31'),"OK",CONCAT(qEnd(DATE('2013-02-01'))," != ",DATE('2013-03-31'))) as qe_b,
IF(qEnd(DATE('2013-03-01')) = DATE('2013-03-31'),"OK",CONCAT(qEnd(DATE('2013-03-01'))," != ",DATE('2013-03-31'))) as qe_c,
IF(qEnd(DATE('2013-04-01')) = DATE('2013-06-30'),"OK",CONCAT(qEnd(DATE('2013-04-01'))," != ",DATE('2013-06-30'))) as qe_d,
IF(qEnd(DATE('2013-05-01')) = DATE('2013-06-30'),"OK",CONCAT(qEnd(DATE('2013-05-01'))," != ",DATE('2013-06-30'))) as qe_e,
IF(qEnd(DATE('2013-06-01')) = DATE('2013-06-30'),"OK",CONCAT(qEnd(DATE('2013-06-01'))," != ",DATE('2013-06-30'))) as qe_f,
IF(qEnd(DATE('2013-07-01')) = DATE('2013-09-30'),"OK",CONCAT(qEnd(DATE('2013-07-01'))," != ",DATE('2013-09-30'))) as qe_g,
IF(qEnd(DATE('2013-08-01')) = DATE('2013-09-30'),"OK",CONCAT(qEnd(DATE('2013-08-01'))," != ",DATE('2013-09-30'))) as qe_h,
IF(qEnd(DATE('2013-09-01')) = DATE('2013-09-30'),"OK",CONCAT(qEnd(DATE('2013-09-01'))," != ",DATE('2013-09-30'))) as qe_i,
IF(qEnd(DATE('2013-10-01')) = DATE('2013-12-31'),"OK",CONCAT(qEnd(DATE('2013-10-01'))," != ",DATE('2013-12-31'))) as qe_j,
IF(qEnd(DATE('2013-11-01')) = DATE('2013-12-31'),"OK",CONCAT(qEnd(DATE('2013-11-01'))," != ",DATE('2013-12-31'))) as qe_k,
IF(qEnd(DATE('2013-12-01')) = DATE('2013-12-31'),"OK",CONCAT(qEnd(DATE('2013-12-01'))," != ",DATE('2013-12-31'))) as qe_l,

IF(qEnd(DATE('2013-01-19')) = DATE('2013-03-31'),"OK",CONCAT(qEnd(DATE('2013-01-19'))," != ",DATE('2013-01-01'))) as qe_ab,
IF(qEnd(DATE('2013-02-19')) = DATE('2013-03-31'),"OK",CONCAT(qEnd(DATE('2013-02-19'))," != ",DATE('2013-03-31'))) as qe_bb,
IF(qEnd(DATE('2013-03-19')) = DATE('2013-03-31'),"OK",CONCAT(qEnd(DATE('2013-03-19'))," != ",DATE('2013-03-31'))) as qe_cb,
IF(qEnd(DATE('2013-04-19')) = DATE('2013-06-30'),"OK",CONCAT(qEnd(DATE('2013-04-19'))," != ",DATE('2013-06-30'))) as qe_db,
IF(qEnd(DATE('2013-05-19')) = DATE('2013-06-30'),"OK",CONCAT(qEnd(DATE('2013-05-19'))," != ",DATE('2013-06-30'))) as qe_eb,
IF(qEnd(DATE('2013-06-19')) = DATE('2013-06-30'),"OK",CONCAT(qEnd(DATE('2013-06-19'))," != ",DATE('2013-06-30'))) as qe_fb,
IF(qEnd(DATE('2013-07-19')) = DATE('2013-09-30'),"OK",CONCAT(qEnd(DATE('2013-07-19'))," != ",DATE('2013-09-30'))) as qe_gb,
IF(qEnd(DATE('2013-08-19')) = DATE('2013-09-30'),"OK",CONCAT(qEnd(DATE('2013-08-19'))," != ",DATE('2013-09-30'))) as qe_hb,
IF(qEnd(DATE('2013-09-19')) = DATE('2013-09-30'),"OK",CONCAT(qEnd(DATE('2013-09-19'))," != ",DATE('2013-09-30'))) as qe_ib,
IF(qEnd(DATE('2013-10-19')) = DATE('2013-12-31'),"OK",CONCAT(qEnd(DATE('2013-10-19'))," != ",DATE('2013-12-31'))) as qe_jb,
IF(qEnd(DATE('2013-11-19')) = DATE('2013-12-31'),"OK",CONCAT(qEnd(DATE('2013-11-19'))," != ",DATE('2013-12-31'))) as qe_kb,
IF(qEnd(DATE('2013-12-19')) = DATE('2013-12-31'),"OK",CONCAT(qEnd(DATE('2013-12-19'))," != ",DATE('2013-12-31'))) as qe_kl