We study a charge pump realized with an elastically deformable quantum dot whose center of mass follows a nonlinear stochastic dynamics. The interplay of noise, nonlinear effects, dissipation and interaction with an external time-dependent driving on the pumped charge is fully analyzed. The results show that the quantum pumping mechanism not only is not destroyed by the force fluctuations, but it becomes stronger when the forcing signal frequency is tuned close to the resonance of the vibrational mode. The robustness of the quantum pump with temperature is also investigated and an exponential decay of the pumped charge is found when the coupling to the vibrational mode is present. Implications of our results for nanoelectromechanical systems are also discussed.