Full classical molecular dynamics (MD) simulations of two-dimensional (2D) infrared-Raman and 2D Raman spectroscopies of liquid water were carried out to elucidate a mode-mode coupling mechanism using a polarizable water model for intermolecular and intramolecular vibrational spectroscopy (POLI2VS). This model is capable of describing both infrared and Raman spectra. Second-order response functions, which consist of one molecular polarizability and two molecular dipole moments for 2D IR-Raman and three molecular polarizabilities for 2D Raman spectroscopies, were calculated using an equilibrium-non-equilibrium hybrid MD approach. The obtained signals were analyzed using a multi-mode Brownian oscillator (BO) model with nonlinear system-bath interactions representing the intramolecular OH stretching, intramolecular HOH bending, hydrogen bonded (HB)-intermolecular librational motion and HB-intermolecular vibrational (translational) motion of liquid water. This model was applied through use of hierarchal Fokker-Planck equations. The qualitative features of the peak profiles in the 2D spectra obtained from the MD simulations are accurately reproduced with the BO model. This indicates that this model captures the essential features of the intermolecular and intramolecular motion. We elucidate the mechanisms governing the 2D signal profiles involving anharmonic mode-mode coupling, the nonlinearities of the polarizability and dipole moment, and the vibrational dephasing processes of liquid water even in the case that the 2D spectral peaks obtained from the MD simulation overlap or are unclear. The mode coupling peaks caused by electrical anharmonic coupling (EAHC) and mechanical anharmonic coupling (MAHC) are observed in all of the 2D spectra. We find that the strength of the MAHC between the OH-stretching and HB-intermolecular vibrational modes is comparable to that between the OH-stretching and HOH bending modes. Moreover, we find that this OH-stretching and HB-intermolecular vibrational coupling should be observed as off-diagonal cross peaks in the 2D spectra.