  We study the possibility of testing quantum gravity (QG) effects with a
type-I seesaw origin of light neutrino mass embedded in a global $U(1)_L$
symmetric framework which also includes dark matter (DM). The spontaneous
breaking of $U(1)_L$ symmetry generates the seesaw scale as well as DM mass
dynamically. The (pseudo) Nambu-Goldstone boson, known as majoron, acquires
non-zero mass due to soft symmetry breaking terms of quadratic type in the
scalar potential, which eventually breaks $U(1)_L$ to its $Z_2$ subgroup. The
spontaneous symmetry breaking, which effectively breaks $Z_2$, leads to the
formation of domain walls (DW), posing a threat to successful cosmology, if
allowed to dominate. As gravity does not respect any global symmetries, we
consider higher dimensional operators suppressed by the scale of QG namely,
$\Lambda_{\rm QG}$ which introduces the required bias leading to DW
annihilation and emission of stochastic gravitational waves (GW) observable at
near future experiments. The same operators also lead to decay of DM bringing
interesting indirect detection aspects. While DM is produced non-thermally via
scalar portal interactions, light majoron can give rise to additional $\Delta
N_{\rm eff}$ within reach of future cosmic microwave background (CMB)
experiments.
