The Hamiltonian of the process of cascaded second harmonic generation is found from Maxwell equations. Inthe double-gap model and under rotating-wave and effective-mass approximations, it is quantized and the generalizedquantum nonlinear Shrodinger equation (GQNSE) is obtained. Tri-photon and quadri-photon bound states are foundbased on general solutions of GQNSE solved via Bethes Ansatz method. Quantum parametric gap soliton (QPGS)solution is constructed consequently, and the existence of the double-gap QPGS is predicted for the first time.