We consider the problem of calculating the nonradiative multiphonon transition rate between two electronic states of an impurity embedded in a condensed-phase environment, where all the nuclear degrees of freedom of the bath are taken in the harmonic approximation, and the two electronic states are coupled to the bath diagonally and off-diagonally. The diagonal coupling term includes displacements of the equilibrium positions of the bath modes, the frequency shifts, and Duschinsky rotations of the bath modes between the two electronic states. We consider two forms of the off-diagonal coupling term-the first assumes that this term is independent of the nuclear degrees of freedom, and thus the coupling between the two diabatic electronic states is taken to be a constant; the second is based on the Born-Oppenheimer method in which the off-diagonal coupling term between the two adiabatic electronic states is taken to be a function of the bath momenta operators. This general model is used to examine the accuracy of several commonly used mixed quantum-classical approximations where the two electronic states are treated quantum mechanically while the bath modes are treated classically. We use the lowest-order perturbation theory to calculate the transition rate between the two electronic states, which is given in terms of the Fourier transform of the off-diagonal coupling-element time correlation function. Following the methodology of Kubo and Toyozawa, we obtain an analytic solution for the fully quantum mechanical time correlation function, and extend our method [S. A. Egorov, E. Rabani and B. J. Berne, J. Chem. Phys. 108, 1407 (1998)] to obtain its mixed quantum-classical counterpart. It is shown that the nonradiative transition rate between the two electronic states calculated using the mixed quantum-classical treatment can deviate by several orders of magnitude from the exact quantum mechanical result. However, the agreement is improved when the classical time propagation of the bath modes is performed with the arithmetic average of the ground- and excited-state nuclear Hamiltonians, and thermal averaging over the initial classical distribution is replaced with the averaging over the corresponding Wigner distribution.