Do we know when the Federation made first contact with the Betazoids?
No. The abandoned opening sequence from
The Undiscovered Country would've had Kirk warning Chekov against playing chess (poker?) against a Betazoid, which would've pegged it before 2293 (and that scene is included in
The Fearful Summons), but canonically there's no evidence locking it down.
As for Trek Lit evidence, the earliest appearance of a Betazoid interacting with the Federation seems to be 2316 according to
Well of Souls -- the year when Rachel Garrett met her Betazoid future husband. So it must've been before then.