We explore the phenomenological consequences of breaking discrete global symmetries in
quantum gravity (QG). We extend a previous scenario where discrete global symmetries are
responsible for scalar dark matter (DM) and domain walls (DWs), to the case of fermionic DM,
considered as a feebly interacting massive particle, which achieves the correct DM relic density
via the freeze-in mechanism. Due to the mixing between DM and the standard model neutrinos,
various indirect DM detection methods can be employed to constrain the QG scale, the scale of
freeze-in, and the reheating temperature simultaneously. Since such QG symmetry breaking leads to
DW annihilation, this may generate the characteristic gravitational wave background, and hence
explain the recent observations of the gravitational wave spectrum by pulsar timing arrays. This
work therefore highlights a tantalizing possibility of probing the effective scale of QG
from observations.