This was actually the subject for a Starfleet Academy movie script back in the ’80s, meant to be Star Trek VI.
A series about Kirk and Spock as Starfleet Academy cadets in the prime timeline might be worth exploring. However, that was all in the alternate Kelvin timeline. In that movie, Kirk and Spock were not exactly the best of friends in their younger days. Abrams already showed young Kirk, Spock, and McCoy as Starfleet cadets.
