May I write a letter to support my girlfriend's application?
Just say no.  No, it's not appropriate.  This is not even close.  She might as well ask her mother to write the LOR.  "She was such a good girl growing up."

It's possible it might not get discovered but if it is, I expect it would (should!) quickly tank her application before most admissions committees.  She needs to find references who can speak to her qualifications without this obvious conflict of interest.