An influence functional approach is used to determine the role of multiphonon processes in the rates of vibrational relaxation. Relaxation is considered to occur between a pair of coupled harmonic oscillators, representing an excited and a receiving mode on a single polyatomic solute, and a collection of independent harmonic oscillators, representing a solvent reservoir. The interaction between the oscillator pair in the solute is arbitrary and left unspecified, while interactions between solute and solvent are taken to be linear in the solute coordinates but quadratic or cubic in the solvent coordinates. The nonlinearities allow vibrational relaxation to occur through multiple excitations of phonons. Transitions rates for such multiphonon processes are derived, as are quantum corrections to the corresponding classical force correlation functions. The quantum correction factors are also shown to emerge directly from certain terms in the real part of the influence functional.