  The measurement of the Higgs quartic coupling modifier between a Higgs boson
pair and a vector boson pair, $\kappa_{2V}$, is expected to be achieved from
vector-boson fusion (VBF) production of a Higgs boson pair. However, this
process involves another unmeasured parameter, the trilinear Higgs
self-coupling modifier $\kappa_\lambda$. A sensitivity analysis should target
both parameters. Since the LHC cannot avoid the gluon fusion pollution, which
becomes severe for non-SM $\kappa_\lambda$, an electron-proton collider is more
appropriate for the comprehensive measurement. In this regard, we study the VBF
production of a Higgs boson pair in the $b\bar{b}b\bar{b}$ final state at the
LHeC and FCC-he. Performing detailed analysis using simulated dataset, we
devise the search strategy specialized at the LHeC and FCC-he and give a
prediction for the sensitivity to both $\kappa_{2V}$ and $\kappa_\lambda$. We
find that the two electron-proton colliders have high potential: the LHeC has
similar exclusion prospects as the HL-LHC; the FCC-he is extremely efficient,
excluding the parameter space outside $\kappa_{2V} \in[0.8, 1.2]$ and
$\kappa_\lambda \in [1, 2.5]$ at 95\% C.L. for the total luminosity of $10$
ab$^{-1}$ and 10\% uncertainty on the background yields.
