Postgres 13

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version
PostgreSQL 13.4 on x86_64-pc-linux-gnu, compiled by gcc (GCC) 8.4.1 20200928 (Red Hat 8.4.1-1), 64-bit

with months(mname) as  
     ( select unnest (mn) 
         from (Select array['Jan','Feb','Mar','Apr','May','Jun','Jul','Aug','Sep','Oct','Nov','Dec']  mn) m 
     )
   , quarters (qtr, qtr_months) as 
     ( values ('Q1', 'Jan, Feb, Mar')
             ,('Q2', 'Apr, May, Jun')
             ,('Q3', 'Jul, Aug, Sep')
             ,('Q4', 'Oct, Nov, Dec')
     ) 
select quarters.qtr "Quarter"
     , quarters.qtr_months "Actual Months in Quarter"
     , qlst.qtr_months  "Months via BETWEEN text compare"
  from quarters 
  left join  
        ( select qtr, string_agg ( mname, ',') qtr_months  
            from (
                   select 'Q1' qtr, mname  
                     from months 
                    where mname between 'Jan' and 'Mar'
                   union all 
                   select 'Q2', mname  
                     from months trtr
                    where mname between 'Apr' and 'Jun ' 
                   union all            
                   select 'Q3', mname  
                     from months 
                    where mname between 'Jul' and 'Sep' 
                   union all           
                   select 'Q4', mname  
                     from months 
                    where mname between 'Oct' and 'Dec' 
                 ) qm
             group by qtr
        ) qlst
      on (qlst.qtr = quarters.qtr);

Quarter	Actual Months in Quarter	Months via BETWEEN text compare
Q1	Jan, Feb, Mar	Jan,Mar,Jun,Jul
Q2	Apr, May, Jun	Jan,Feb,Apr,Jun,Jul,Aug,Dec
Q3	Jul, Aug, Sep	Mar,May,Jun,Jul,Sep,Oct,Nov
Q4	Oct, Nov, Dec	null