SELECT count(`Заказчик`) FROM `posts` WHERE Заказчик LIKE 'Название"'в поле.
DELIMITER $$
CREATE TRIGGER tgr_Posts_ins
AFTER INSERT
ON Posts FOR EACH ROW
BEGIN
SELECT count(1) INTO @countObj from Posts where ObjName = NEW.ObjName;
UPDATE Objects set PostsAmount = @countObj where ObjName = NEW.ObjName;
END$$
DELIMITER;