The Turbulence Damping Function is part of the Flight Control System and not the Auto Flight System. It is normally on and not dependant on an AP being engaged.
There is no autopilot turbulence mode and the Turbulence Damping Function is somewhat different from the TURB autopilot mode of previous non-FBW Airbus. That would disengage pitch mode which would revert to pitch hold while the lateral mode would be retained and THS trim speed reduced.