  Motivated by the recently synthesized insulating nickelate Ni$_2$Mo$_3$O$_8$,
which has been reported to have an unusual non-collinear magnetic order of
Ni$^{2+}$ $S=1$ moments with a nontrivial angle between adjacent spins, we
construct an effective spin-1 model on the honeycomb lattice, with the exchange
parameters determined with the help of first principles electronic structure
calculations. The resulting bilinear-biquadratic model, supplemented with the
realistic crystal-field induced anisotropy, favors the collinear N\'eel state.
We find that the crucial key to explaining the observed noncollinear spin
structure is the inclusion of the Dzyaloshinskii--Moriya (DM) interaction
between the neighboring spins. By performing the variational mean-field and
linear spin-wave theory (LSWT) calculations, we determine that a realistic
value of the DM interaction $D\approx 2.78$ meV is sufficient to quantitatively
explain the observed angle between the neighboring spins. We furthermore
compute the spectrum of magnetic excitations within the LSWT and random-phase
approximation (RPA) which should be compared to future inelastic neutron
measurements.
