We investigate the intramolecular quantum phonon modes effect in the non-adiabatic electron tunneling through a redox molecule with a single electron level in a symmetric contact in the electrolyte solution at room temperature. The Coulomb blockade effect is taken into account within the infinite Coulomb repulsion limit. The equilibrium electrode potential is defined and the tunnel current/overvoltage dependence is studied. The spectral function is calculated and its physical meaning is discussed. The differential conductance/bias voltage and the tunnel current/bias voltage correlations are considered. It is shown that these correlations depend strongly on the value of the overvoltage and the extent of Debye screening of the electric field in the tunnel gap. The effect of the intramolecular quantum phonon mode on rectification is studied. It is shown that the account of this effect can lead to significant increase of the rectification ratio. (C) 2010 Elsevier B.V. AU rights reserved.