If you mean commercially produced Star Trek themed first person shooters, my answer would be a "maybe", although the likelihood for it to happen relatively concurrently with the next movie wouldn't be too great as there is nothing on the horizon at the moment (that I'm aware of). If you mean the same thing but Voyager themed, I'll let Alfred do the talking.
Ironic that one of the "worst" Star Trek series got one of the best video games based on the property. Good graphics for the time and a cool story to boot. There was a 3rd person shooter based on the the new crew not to long ago on PS3. I never played it though.
Star Trek: The Video Game, came out for PC, Xbox 360 and PS3 in 2013. It was okay-ish, albeit ridden with AI issues if you're playing solo. Played cooperatively, it's a perfectly serviceable TPS. That game is the definition of the old adage "good, for a Star Trek game".
Loved the Elite Force games. And yeah, the Star Trek game based on the new movies (rather, between the new movies) was ok. Agree with SPCTRE...largely A.I. issues... more A than I. However, heavy stun is your best friend in the game, because it will help deal with most enemies fairly quickly... especially if you're going for a non-lethal achievement/trophy.
I've been playing these games recently, tried some editing of them, in particular the E deck NX01 I released it a couple of weeks ago.