We use a symmetry-based structural analysis combined with an electronic descriptor for bond covalency to explain the origin of the second-order nonlinear optical response (second harmonic generation, SHG) in noncentrosymmetric nonpolar ATeMoO(6) compounds (where A = Mg, Zn, or Cd). We show that the SHG response has a complex dependence on the asymmetric geometry of the AO(6) and AO(4) functional units and the orbital character at the valence band edge, which we are able to distinguish using an A-O bond covalency descriptor. The degree of covalency between the divalent A-site cation and the oxygen ligands dominates over the geometric contributions to the SHG arising from the acentric polyhedra, and this can be understood from considerations of the local static charge density distribution. The use of a local dipole model for the polyhedral moieties (AO(4)/AO(6), MoO4, and TeO4) can account for a nonzero SHG response, even though the materials exhibit nonpolar structures; however, it is insufficient to explain the change in the magnitude of the SHG response upon A-cation substitution. The atomic scale and electronic structure understanding of the macroscopic SHG behavior is then used to identify hypothetical HgTeMoO6 as a candidate telluromolybdate with an enhanced nonlinear optical response.