  We use quantum Monte Carlo simulations to study a dimer-diluted $S=1/2$
Heisenberg model on a bilayer square lattice with intralayer interaction
$J_{1}$ and interlayer interaction $J_{2}$. Below the classical percolation
threshold $p_c$, the system has three phases reachable by tuning the
interaction ratio $g=J_{2}/J_{1}$: a N$\acute{e}$el ordered phase, a gapless
quantum glass phase, and a gapped quantum paramagnetic phase. We present the
ground-state phase diagram in the plane of dilution $p$ and interaction ratio
$g$. The quantum glass phase is certified to be of the gapless Mott glass type,
having a uniform susceptibility vanishing at zero temperature $T$ and following
a stretched exponential form at $T>0$; $\chi_u \sim \exp(-b/T^{\alpha})$ with
$\alpha < 1$. At the phase transition point from N$\acute{e}$el ordered to Mott
glass, we find that the critical exponents are different from those of the
clean system described by the standard $O(3)$ universality class in 2+1
dimensions.
