Classical non-perturbative simulations of open quantum systems dynamics face several scalability problems, namely, exponential scaling as a function of either the time length of the simulation or the size of the open system. In this work, we propose a quantum computational method which we term as Quantum Time Evolving Density operator with Orthogonal Polynomials Algorithm (Q-TEDOPA), based on the known TEDOPA technique, avoiding any exponential scaling in simulations of non-perturbative dynamics of open quantum systems on a quantum computer. By performing an exact transformation of the Hamiltonian, we obtain only local nearest-neighbour interactions, making this algorithm suitable to be implemented in the current Noisy-Intermediate Scale Quantum (NISQ) devices. We show how to implement the Q-TEDOPA using an IBM quantum processor by simulating the exciton transport between two light-harvesting molecules in the regime of moderate coupling strength to a non-Markovian harmonic oscillator environment. Applications of the Q-TEDOPA span problems which can not be solved by perturbation techniques belonging to different areas, such as dynamics of quantum biological systems and strongly correlated condensed matter systems.