Depends mainly on how you identify Quarter / year values (e.g. according to business / fiscal calendar)?
Then you can convert those values to your desired display format. For example:
CAST(td_quarter_of_year(theDate) AS VARCHAR(1))||'Q'||CAST(CAST(theDate AS FORMAT 'YY') AS VARCHAR(2))
You may want create a SQL UDF to do this.
Or build a calendar table with one row per date that includes various display formats, and join to that.