There is definitely no option in the FH built-in Export > GEDCOM File... command to alter date formats.
As I said, the general reason is that most products only recognise GEDCOM Dates.
If non-GEDCOM Dates are imported to FH they become Date Phrases that need converting by a Plugin, which is a pain.
I think Quarter Dates are a largely UK phenomenon and so did not figure in the GEDCOM Specification.
It would be fairly easy to write a script for a Plugin or Notepad++ or many other tools to automate the translation from Between Date to Quarter Date.
There are only four relative rare cases to deal with if the 1st year can be wildcarded:
DATE BTW JAN 1867 AND MAR 1867 => DATE Q1 1867 i.e. replace DATE BTW JAN **** AND MAR with DATE Q1
DATE BTW APR 1867 AND JUN 1867 => DATE Q2 1867
DATE BTW JUL 1867 AND SEP 1867 => DATE Q3 1867
DATE BTW OCT 1867 AND DEC 1867 => DATE Q4 1867
There is a tiny risk that the 1st year might not be the same as the last year, but I suspect you could live with that.