A.F. were the initials of an "acquaintance" of Jean-Luc Picard.

Picard later attributed his failure in Organic Chemistry to his involvement with A.F., whose initials he carved into Boothby's prized elm tree near the parade grounds at Starfleet Academy. (TNG: "The Game")

This character was only mentioned in dialogue.
In the script for the "The Game", the initials were S.P.