In a series of two papers, we aim to develop an efficient and accurate approach to simulate the nonlinear spectra of the molecule-cavity hybrid system under the collective coupling regime. In Paper I (this work), we will focus on introducing two theoretical techniques that allow one to significantly reduce the computational cost of simulating the quantum dynamics of the HTC system with various types of disorders. We will focus on the linear spectra simulation. In Paper II, we will outline the theoretical details of stimulating 2DES for the polariton system under the collective coupling regime, with new computational techniques for focusing the algorithm of the PLDM simulation.