  We study vanadium spinels $A$V$_2$O$_4$ ($A$ = Cd, Mg) in pulsed magnetic
fields up to 65 T. A jump in magnetization at $\mu_0 H \approx$ 40 T is
observed in the single-crystal MgV$_2$O$_4$, indicating a field induced quantum
phase transition between two distinct magnetic orders. In the multiferroic
CdV$_2$O$_4$, the field-induced transition is accompanied by a suppression of
the electric polarization. By modeling the magnetic properties in the presence
of strong spin-orbit coupling characteristic of vanadium spinels, we show that
both features of the field-induced transition can be successfully explained by
including the effects of the local trigonal crystal field.
