** This code is tested on Oracle Database. For any syntax related error, you can replace according to you database.
Also, update your column name,date,timestamp,timezone and table name accordingly.
The sample query is like this ...
select temp.quarter,avg(resolve_time)
from
(
select resolve_time,
CASE
WHEN (close_date between TO_DATE('2019-APR-01 00:00','YYYY-MON-DD HH24:MI','NLS_DATE_LANGUAGE=AMERICAN') and TO_DATE('2019-JUN-30 23:59','YYYY-MON-DD HH24:MI','NLS_DATE_LANGUAGE=AMERICAN')) THEN 1
WHEN (close_date between TO_DATE('2019-JUL-01 00:00','YYYY-MON-DD HH24:MI','NLS_DATE_LANGUAGE=AMERICAN') and TO_DATE('2019-SEP-30 23:59','YYYY-MON-DD HH24:MI','NLS_DATE_LANGUAGE=AMERICAN')) THEN 2
WHEN (close_date between TO_DATE('2019-OCT-01 00:00','YYYY-MON-DD HH24:MI','NLS_DATE_LANGUAGE=AMERICAN') and TO_DATE('2019-DEC-31 23:59','YYYY-MON-DD HH24:MI','NLS_DATE_LANGUAGE=AMERICAN')) THEN 3
WHEN (close_date between TO_DATE('2020-JAN-01 00:00','YYYY-MON-DD HH24:MI','NLS_DATE_LANGUAGE=AMERICAN') and TO_DATE('2020-MAR-31 23:59','YYYY-MON-DD HH24:MI','NLS_DATE_LANGUAGE=AMERICAN')) THEN 4
ELSE 0 END quarter
from TestStack
) temp
group by temp.quarter;