We investigate theoretically four-wave mixing (FWM) response and optical bistability (OB) in a hybrid nanosystem composed of a metal nanoparticle (MNP) and a semiconductor quantum dot (SQD) coupled to a nanomechanical resonator (NR). It is shown that the FWM signal is enhanced by more than three orders of magnitude as compared to that of the system without exciton-phonon interaction, and the FWM signal can also be suppressed significantly and broadened due to the exciton-plasmon interaction. As the MNP couples strongly with the SQD, the bistable FWM response can be achieved by adjusting the SQD-MNP distance and the pumping intensity. For a given pumping constant and a fixed SQD-MNP distance, the enhanced exciton-phonon interaction can promote the occurrence of bistability. Our findings not only present a feasible way to detect the spacing between two nanoparticles, but also hold promise for developing quantum switches and nanoscale rulers.