Como obtenho a primeira data de um trimestre no MySQL?

tente isso:

Para obter a data de início do trimestre atual, use isto:

  SELECT  MAKEDATE(YEAR(CURDATE()), 1) + INTERVAL QUARTER(CURDATE()) QUARTER 
                                       - INTERVAL    1 QUARTER

Então sua consulta seria:

SELECT
 QUARTER(r.callDate) AS quar,
 YEAR(r.callDate) AS ryear,
 MAKEDATE(YEAR(r.callDate), 1) + INTERVAL QUARTER(r.callDate) QUARTER -
  INTERVAL 1 QUARTER  AS scoreDateStart,
 (SELECT DATE (DATE_SUB( DATE_ADD( CONCAT( YEAR( r.callDate ), '-01-01'), 
 INTERVAL QUARTER(r.callDate) QUARTER ), INTERVAL 1 DAY))) AS scoreDateEnd,
 group_concat(DISTINCT(r.resultId) separator ', ') AS resultIds
FROM results AS r
GROUP BY  quar, ryear
ORDER BY quar;