Will 12" studs work for a TV mount designed for 16" stud mounting?
You haven't told us what mount you're referring to, but most aren't designed for specific stud centers. Instead, they have a range of available mounting area (and many will accommodate 24" centers as well). It's very likely that your mount will work just fine on 12" centers. 

You should be able to put a tape measure on the mount and verify. Can you position two bolts 12" apart? If so, the only question is whether you'll be able to mount the TV exactly where you want it, or if you'll have to shift to the side a bit.