Extension of trailing edge flaps will produce (at least) two competing effects:
Effect 1. The CP will move aft, causing a nose down pitching moment.
You may wish to argue that with CP ahead of CG the result will actually be a reduction in the nose up pitching moment (about the CG). But if the aircraft is originally in a trimmed condition, the result of the aft CP shift would be a nose down pitching moment.
Effect 2. If the tailplane is within the downwash from the mainplanes,the flap extension will cause the downwash over the tailplane to increase. This will cause a nose up pitching moment.
The overall result will depend upon the relative magnitudes of these two effects.