  We report a method for determining the spatial dependence of the magnetic
exchange coupling, $dJ/dr$, from magnetostriction measurements of a quantum
magnet. The organic Ni $S = 1$ system NiCl$_2$-4SC(NH$_2$)$_2$ exhibits lattice
distortions in response to field-induced canted antiferromagnetism between
$H_{c1} = 2.1$ T and $H_{c2} = 12.6$ T. We are able to model the
magnetostriction in terms of uniaxial stress on the sample created by magnetic
interactions between neighboring Ni atoms along the c-axis. The uniaxial strain
is equal to $(1/E)dJ_c/dx_c < S_{\bf r} \cdot S_{{\bf r}+ {\bf e}_c} >$, where
$E$, $J_c$, $x_c$ and ${\bf e}_c$ are the Young's modulus, the nearest neighbor
(NN) exchange coupling, the variable lattice parameter, and the relative vector
between NN sites along the c-axis. We present magnetostriction data taken at 25
mK together with Quantum Monte Carlo calculations of the NN spin-spin
correlation function that are in excellent agreement with each other. We have
also measured Young's modulus using resonant ultrasound, and we can thus
extract $dJ_c/dx_c = 2.5$ K/$\AA$, yielding a total change in $J_c$ between
$H_{c1}$ and $H_{c2}$ of 5.5 mK or 0.25% in response to an 0.022% change in
length of the sample.
