Изтрийте ред с Mysql Trigger

Във вашата клауза where променете new.id на old.id

-- Trigger DDL Statements
DELIMITER $$

USE `db_dhruniversity`$$

CREATE
TRIGGER `db_dhruniversity`.`ai_delete_course`
AFTER DELETE ON `jos_jquarks_quizzes`
FOR EACH ROW
BEGIN    
DELETE FROM jos_jquarks_courses
WHERE (quiz_id = old.id); 
END$$