Nuclear quadrupole resonance techniques have been used to investigate magnetic interactions in the bromates and iodates of certain $3d$ transition metal ions. In particular, for Ni${(\mathrm{I}{\mathrm{O}}_{3})}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}2${\mathrm{H}}_{2}$O there is an abrupt disappearance of the ${\mathrm{I}}^{127}(\ifmmode\pm\else\textpm\fi{}\frac{3}{2}\ensuremath{\leftrightarrow}\ifmmode\pm\else\textpm\fi{}\frac{1}{2})$ transition at 3.08\ifmmode^\circ\else\textdegree\fi{}K; at lower temperatures there is a large, temperature-dependent splitting. This behavior is attributed to the combined effects of an antiferromagnetic ordering of the ${\mathrm{Ni}}^{++}$ electron spins and a hfs interaction of the nonlocalized ${\mathrm{Ni}}^{++}$ spin magnetization with the ${\mathrm{I}}^{127}$ nuclear magnetic moment. Measurements of the pure quadrupole transition frequencies above, and of their temperature-dependent splittings below, the N\'eel temperature yield the quadrupole coupling constant, the asymmetry of the electric field gradient (EFG) tensor, the magnitude and orientation of the internal magnetic field relative to the principal axes of the EFG tensor, and the magnitude and orientation of the internal magnetic field relative to the principal axes of the EFG tensor, and the sublattice magnetization as a function of temperature. Qualitative experiments at 1.3\ifmmode^\circ\else\textdegree\fi{}K indicate ${{T}_{1}}^{127}\ensuremath{\ll}100$ sec and ${{T}_{2}}^{127}\ensuremath{\sim}{10}^{\ensuremath{-}6}$ sec. Measurements of the rf and dc magnetic susceptibility have been made and are consistent with these conclusions and, in addition, indicate the sudden onset of a spontaneous ferromagnetic moment as the temperature is lowered below ${T}_{n}$. A possible isotope effect on the ${T}_{n}$ of Ni(I${\mathrm{O}}_{3}$)\ifmmode\cdot\else\textperiodcentered\fi{}2${\mathrm{H}}_{2}$O was looked for but none was found. Cupric iodate and the bromates give no evidence of magnetic ordering above 1.3\ifmmode^\circ\else\textdegree\fi{}K.