The radially dependent quadrupole antishielding factors $\ensuremath{\gamma}(r)$ and $\ensuremath{\beta}(r)$ for ${\mathrm{Be}}^{2+}$, ${\mathrm{Mg}}^{2+}$, and ${\mathrm{Zn}}^{2+}$ are calculated by using the nonorthogonal-Hartree-Fock perturbation theory (NHFPT) of Dalgarno. These results are important in view of the recent theory of Lodge on the electric-field gradient (EFG) in metals, which shows the dependence of EFG on both $\ensuremath{\gamma}(r)$ and $\ensuremath{\beta}(r)$. While in the literature, calculations of $\ensuremath{\gamma}(r)$ are available for some ions (other than ${\mathrm{Be}}^{2+}$, ${\mathrm{Mg}}^{2+}$, and ${\mathrm{Zn}}^{2+}$), those of $\ensuremath{\beta}(r)$ are lacking to our knowledge. It is this deficiency which is being made up in the present paper by the calculations of $\ensuremath{\beta}(r)$ and $\ensuremath{\gamma}(r)$ for some of the divalent ions.