We have found experimentally the IR-induced second harmonic generation (SHG) in glasses possessing different degrees of electron-phonon interactions. For the investigations, we have chosen As1-xBixTe3-CaBr2-PbBr2 (0 < x < 1) glasses. General formalism is based on consideration of fifth-order nonlinear optical susceptibility. The effect is observed in the middle IR region (spectral range 0.92-10.5 mum) where the value of the electronic energy gap is commensurable to the energies of actual quasi-phonons participating in the anharmonic (non-centrosymmetric) electron-phonon interactions. Varying the As/Bi ratio allows us to operate by the degree of electron-phonon anharmonicity in a wide spectral range. The second harmonic generation (SHG) output signal shows a correlation with IR-induced anharmonic phonon modes within the 1.5-4.8 mum spectral range. A maximum value of SHG is achieved at pump-probe delaying times of about 12.5-20 ps, which are typical for relaxation of the anharmonic electron-quasi-phonon subsystem. The maximally achieved value of the phonon-assisted optical susceptibility was about 6 x 10(-38) m(4)/V-4. The SHG signal was saturated for the IR pump power densities of about 1.73 GW/cm(2), corresponding to output SHG signals of about 9.8 x 10(-4) with respect to the fundamental ones. By varying the degree of electron-phonon anharmonicity and changing content of glasses, it was unambiguously shown that the IR-induced SHG signal correlates well with changes of oscillator strengths of IR-induced anharmonic phonon modes.