  The quartic Higgs-gauge-boson coupling $g_{hhWW}$ is sensitive to the
electroweak symmetry breaking mechanism, however, it is challenging to be
measured at the large hadron collider. We show that the coupling can be well
probed at future lepton colliders through di-Higgs boson production via the
vector boson fusion channel. We perform a detailed simulation of
$\ell^+\ell^-\to \nu\bar\nu hh\to 4b\,+ \!\!\not{\!\! E}_T$ with parton
showering effects at $e^+e^-$, $\mu^+\mu^-$ and $e^-\mu^+$ colliders. We find
that the regions of $g_{hhWW}/g_{hhWW}^{\rm SM}<0.86$ and
$g_{hhWW}/g_{hhWW}^{\rm SM}>1.32$ can be discovered at the $5\sigma$ confidence
level at the 10 TeV $\mu^+ \mu^-$ collider with an integrated luminosity of 3
ab$^{-1}$.
