(Dated: July 17, 2017) We calculate the electric charge current flowing through a vibrating molecular nanojunction, which is driven by an ac voltage, in its regime of nonlinear oscillations. Without loss of generality, we model the junction by a vibrating molecule which is doubly clamped to two metallic leads which are biased by time-periodic ac voltages. Dressed-electron tunneling between the leads and the molecule drives the mechanical degree of freedom out of equilibrium. In the deep quantum regime, where only a few vibrational quanta are excited, the formation of coherent vibrational resonances affects the dressed-electron tunneling. In turn, back action modifies the electronic ac current passing through the junction. The concert of nonlinear vibrations and ac driving induces quantum transport currents which are antiresonant to the applied ac voltage. Quantum back action on the flowing nonequilibriun current allows us to obtain rather sharp spectroscopic information on the population of the mechanical vibrational states.