We present an investigation of the effect of randomizing exchange coupling strengths in the S = 1/2 square lattice quasi-two-dimensional quantum Heisenberg antiferromagnet (QHAF) (QuinH)₂Cu(Cl_xBr_1-x)₄ ⋅ 2H₂O (QuinH = Quinolinium, C₉H₈N⁺), with 0 ≤ x ≤ 1. Pulsed-field magnetization measurements allow us to estimate an effective in-plane exchange strength J in a regime where exchange fosters short-range order, while the temperature T_N at which long-range order (LRO) occurs is found using muon-spin relaxation, allowing us to construct a phase diagram for the series. We evaluate the effectiveness of disorder in suppressing T_N and the ordered moment size, and we find an extended disordered phase in the region 0.4 ≲ x ≲ 0.8 where no magnetic order occurs. The observed critical substitution levels are accounted for by an energetics-based competition between different local magnetic orders. Furthermore, we demonstrate experimentally that the ground-state disorder is driven by quantum effects of the exchange randomness, which is a feature that has been predicted theoretically and has implications for other disordered quasi-two-dimensional QHAFs.