  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 the
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.
