Electron transport through a quantum dot or single molecule coupled to a quantum oscillator is studied by the Keldysh nonequilibrium Green's function formalism to obtain insight into the quantum dynamics of the electronic and oscillator degrees of freedom. We tune the electronic level of the quantum dot by a gate voltage, where the leads are kept at zero temperature. Due to the nonequilibrium distribution of the electrons in the quantum dot, the spectral function becomes a function of the gate voltage. Novel spectral features are identified for the ground and excited states of nanomechanical oscillators that can be used to enhance the measurement sensitivity.