SET @startAt = "2022-12-01 00:00:00";
SET @endAt = "2023-02-01 00:00:00";

SELECT
	*
FROM customer_mother cm
	
WHERE
	(YEAR (@startAt)= YEAR (@endAt) AND DATE_FORMAT(birthday, "%m%d" ) BETWEEN DATE_FORMAT(@startAt, "%m%d" ) AND DATE_FORMAT(@endAt, "%m%d" ))
	OR (
		YEAR (@startAt)< YEAR (@endAt) 
		AND ((DATE_FORMAT(@startAt, "%m%d" )<= DATE_FORMAT(@endAt, "%m%d" )) 
			OR (
				DATE_FORMAT(@startAt, "%m%d" )> DATE_FORMAT(@endAt, "%m%d" ) 
				AND (
						 (DATE_FORMAT(birthday, "%m%d" ) BETWEEN DATE_FORMAT(@startAt, "%m%d" ) AND '1231' ) 
					OR (DATE_FORMAT(birthday, "%m%d" ) BETWEEN '0101' AND DATE_FORMAT(@endAt, "%m%d" ))
				)
			)
		)
	)