Biological systems, such as green sulphur bacteria, have been found to have exceptional light-harvesting capabilities. A key component of this process occurs in the Fenna-Matthews-Olson (FMO) complex, where energy is transferred with near-perfect efficiency to the reaction site. Despite being a system open to its environment, it has been suggested that observed quantum effects play an important role in achieving such high efficiency. In this work, we look to simulate the inter-molecular site dynamics of the FMO complex using a digital quantum computer. By modeling the vibrational phonon modes as harmonic oscillators, we verify our algorithm recovers site statistics of electron-phonon systems with a strong agreement with simulations on classical devices. Subsequently, we apply our model to simulate the full FMO complex and compare our results to the most recent experimental observations. We find that achieving higher accuracy on such simulations is limited by the number and quality of available qubits and suggest ways this can be overcome in the near future. *BJ acknowledges support from the EPSRC National Quantum Technology Hub in Networked Quantum Information Technology (EP/M013243/1) and the EPSRC Hub in Quantum Computing and Simulation (EP/T001062/1).