  We present an investigation of the effect of randomizing exchange strengths
in the $S=1/2$ square lattice quasi-two-dimensional quantum Heisenberg
antiferromagnet (QuinH)$_2$Cu(Cl$_{x}$Br$_{1-x}$)$_{4}\cdot$2H$_2$O
(QuinH$=$Quinolinium, C$_9$H$_8$N$^+$), with $0\leq x \leq 1$. 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 $T_{\mathrm{N}}$ 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
$T_{\mathrm{N}}$ and the ordered moment size and find an extended disordered
phase in the region $0.4 \lesssim x \lesssim 0.8$ where no magnetic order
occurs, driven by quantum effects of the exchange randomness.
