The moon does rotate on its axis, just enough to stay in tidal lock with the Earth. So I suppose you’re really asking if there are moons that are not in 1:1 tidal lock with their planets.
Mercury’s spin has a 3:2 resonance with its orbit, so I bet that might happen for some planetary moons as well. I can’t do the math, but I believe that 1:1 is more typical for more cirular orbits, whereas other resonances are associated with more elliptical ones.
The other resonances aren’t resonances at all unless you’ve got an elliptical orbit. And Mercury’s orbit is the most elliptical of any of the rockballs or gasballs, so it’s not surprising that it’s got a resonant lock.
That said, while it wouldn’t be too surprising to find a moon in a resonant lock with its planet, such a situation is not known in our Solar System (though of course, there might be some wholly unremarkable chunk of ice around one of the gas giants that we’ve just never noticed).
Minor nitpick. The major moons of the gas giants are thought to have formed in place, with the exception of Triton. Of the smaller satellites, some formed in place, some were captured and others are thought to be the relics of collisions.