We theoretically investigate the magnetoelastic coupling (MEC) and its effect on magnon transport in two-dimensional antiferromagnets with a honeycomb lattice. MEC coefficients along with magnetic exchange parameters and spring constants are computed for monolayers of transition-metal trichalcogenides with Neel magnetic order (MnPS3 and VPS3) and zigzag order (CrSiTe3, NiPS3, and NiPSe3) by ab initio calculations. Using these parameters, we predict that the spin-Nernst coefficient is significantly enhanced due to magnetoelastic coupling. Our study shows that although Dzyaloshinskii-Moriya interaction can produce spin-Nernst effect in these materials, other mechanisms such as magnon-phonon coupling should be taken into account. We also demonstrate that the magnetic anisotropy is an important factor for control of magnon-phonon hybridization and enhancement of the Berry curvature and thus the spin-Nernst coefficient. Our results pave the way toward gate tunable spin current generation in two-dimensional magnets by spin-Nernst effect via electric field modulation of MEC and anisotropy.