When a mount and a rider move together, they are both subject to AoOs.
I *think* you're asking about the following situation:
Mount and rider are in melee combat within reach of an opponent.
The rules state "Your mount acts on your initiative count as you direct it."
We'll assume either a combat-trained mount and/or the necessary ride/handle animal checks are made.
The mount does a withdraw maneuver while the rider makes a single strike.
The rules state "If your mount moves more than 5 feet, you can only make a single melee attack."
The mount would not provoke for leaving the first square that it started in, but the rider would (as the rider is NOT performing a withdraw).