Can you install travertine tiles over plywood sub floor?

It depends on how thick the subfloor is. Travertine, marble, etc, are *heavy* and most modern buildings are inadequate - one sheet of ply over joists is not going to prevent the floor from bouncing as you walk over it. The more the floor moves - especially if we're talking a new house that hasn't settled yet - the more the tiles shift. This leads to grout cracking and, in the worst cases, tiles working loose or breaking when you step on them.

Check the thickness of the subfloor 'in toto' - you want at least a 1-1/2" depth to carry a load of heavy stone tile. If you need to, add plywood or (better yet) cement board (doesn't swell or shrink according to humidity) to make up the difference.