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.
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