Does the center of mass for an airplane have to be over the wings? I am designing a model aircraft that will have 4 propellers in a square configuration (think like a quadcopter) so that it can take off and move around like a quadcopter, then transition into plane flight like a VTOL aircraft. 

For operating in quadcopter mode, it would be ideal for the center of mass to be in the center of the airframe (behind the main wing) for motor efficiency and agility reasons. If the CG is over the wing the front two motors will have to work harder than the back two, and while the PID controls will stabilize for this, it will decrease throttle headroom.
No.

One relatively trivial example is a tandem wing (or a canard with a very large forward surface, if you will); the center of mass will be between the lifting surfaces, positioned so that the forward surface flies at a higher coefficient of lift than the rear (to give static longitudinal stability).

If you build your model with the motors on short arms, so that they're a bit above the wings when in quadcopter mode, you may be able to manage the CoM shift so that you automatically get the balance right for forward flight in wing lift.