Sounds like a bad idea. I'm guessing you have vertical dropouts, which makes riding it fixed nigh-impossible, unless you use an eccentric hub or magic gear ratio. Magic gear is out for track racing, as people generally change gearing to suit the event. As for eccentric hubs on the track, idk.

Furthermore, it will still have road geo, which you may or may not like for the track (and lower bottom bracket, which could potentially be problematic)