ZrSiS has recently gained attention due to its unusual electronic properties: nearly perfect electron-hole compensation, large, anisotropic magneto-resistance, multiple Dirac nodes near the Fermi level, and an extremely large range of linear dispersion of up to ∼ 2 eV. We have carried out a series of high pressure electrical resistivity measurements on single crystals of ZrSiS. Shubnikov-de Haas measurements show two distinct oscillation frequencies. For the smaller orbit, we observe a change in the phase of ∼0.5, which occurs between 0.16 − 0.5 GPa. This change in phase is accompanied by an abrupt decrease of the cross-sectional area of this Fermi surface. We attribute this change in phase to a possible topological quantum phase transition. The phase of the larger orbit exhibits a Berry phase of π and remains roughly constant up to ∼2.3 GPa. Resistivity measurements to higher pressures show no evidence for pressure-induced superconductivity to at least ∼ 20 GPa.