  TmVO$_{4}$ exhibits ferroquadrupolar order of the Tm 4f electronic orbitals
at low temperatures, and is a model system for Ising nematicity that can be
tuned continuously to a quantum phase transition via magnetic fields along the
$c$-axis. Here we present $^{51}$V nuclear magnetic resonance data in magnetic
fields perpendicular to the $c$-axis in a single crystal that has been
carefully cut by a plasma focused ion beam to an ellipsoidal shape to minimize
the inhomogeneity of the internal demagnetization field. The resulting dramatic
increase in spectral resolution enabled us to resolve the anisotropy of the
electric field gradient and to measure the magnetic and quadrupolar relaxation
channels separately. Perpendicular magnetic fields nominally do not couple to
the low energy degrees of freedom, but we find a significant nonlinear
contribution for sufficiently large fields that give rise to a rich phase
diagram. The in-plane magnetic field can act either as an effective transverse
or longitudinal field to the Ising nematic order, depending on the orientation
relative to the principle axes of the quadrupole order, and leads to a marked
in-plane anisotropy in both relaxation channels. We find that the small
in-plane transverse fields initially enhance the ferroquadrupolar ordering
temperature but eventually suppress the long-range order. We tentatively
ascribe this behavior to the competing effects of field-induced mixing of
higher energy crystal field states and the destabilizing effects of
field-induced quantum fluctuations.
