The experimental observation of quantum phenomena in mechanical degrees of freedom is difficult, as the systems become linear toward low energies and the quantum limit, and thus reside in the correspondence limit. Here we investigate how to access quantum phenomena in flexural nanomechanical systems which are strongly deflected by a voltage. Near a metastable point one can achieve a significant nonlinearity in the electromechanical potential at the scale of zero-point energy. The system can then escape from the metastable state via macroscopic quantum tunneling (MQT). We consider two model systems suspended atop a voltage gate, namely, a graphene sheet and a carbon nanotube. We find that the experimental demonstration of the phenomenon is currently possible but demanding, since the MQT crossover temperatures fall in the milli-Kelvin range. A carbon nanotube is suggested as the most promising system.