The FH display of Quarter Dates is just a superficial cosmetic display option.
See Tools > Preferences > General tab, Use quarter date display format.
If that is not ticked then the standard GEDCOM Date Range format is displayed such as "bet Jul 1867 and Sep 1867".
Regardless, of that tick option the actual Project GEDCOM file always uses the Date Range format.
So, any exported GEDCOM will also use that Date Range format.
Any product importing a GEDCOM will expect that format and won't recognize "Q3 1867" and I think that includes FH.
Why do you want the GEDCOM to use the Quarter Date format?