All life on Earth relies on the ability of photosynthetic organisms to efficiently harvest and trap energy from sunlight. Acting as a molecular wire, a protein-pigment complex (PPC) known as the Fenna-Matthews-Olson (FMO) complex found in green sulfur bacteria mediates the transfer of photo-excitation energy between the photosynthetic antennae complex, where energy is harvested, and the reaction centre, where it is trapped. The fine balance between intra-system and system-bath couplings present in the FMO complex allows it to perform unidirectional excitonic energy transfer (EET) with an almost 100% quantum yield. Using coherent theories, quantum dynamical treatment of the bath-influenced EET process can simulate, in silico, coherence effects that have been observed experimentally. The celebrated hierarchical equations of motion (HEOM) approach accurately describes EET dynamics and successfully accounts for non-equilibrium and non-Markovian effects. Though exact, with very few assumptions made about the dynamics or state of the system, HEOM is computationally very expensive for large systems. This motivates the use of a quantum master equation, such as the Lindblad equation formed under the Born-Markov approximation, as an alternative and cheaper description of EET. One such Lindblad model, in agreement with the HEOM approach and experiment, is particularly effective in describing the EET dynamics in the FMO complex despite the minimal computational cost. This work represents a master's thesis submitted in partial fulfilment of the requirements for the Honours degree of MSci Chemistry at the University of Bristol, authored by Joseph W. Abbott in 2020. A supporting Python package complementary to this thesis can be found on GitHub at https://github.com/jwa7/quantum_HEOM.