The temperature-dependent first-order quantum correction contributions to the two-dimensional vibrational response functions, that are linearly proportional to (h) over bar, are theoretically investigated. By carrying out the semiclassical expansion of the nonlinear response function in the limit of weak anharmonicity of the potential-energy surface, seventy nine quantum correction terms are obtained. Although it is the cubic anharmonic couplings that were found to be important at the level of classical calculation of the associated nonlinear response functions, it is found that the quartic anharmonic terms can play a role as the quantum contributions. For a three-Brownian-oscillator model system, various numerical calculations of the 2D (two-dimensional) spectra are carried out to show how these quantum correction terms depend on temperature as well as on relative amplitudes of the quartic and cubic anharmonic coupling constants.