  The quartic gauge boson couplings that identify the strengths of the gauge
boson self-interactions are exactly determined by the non-Abelian gauge nature
of the Standard Model. The examination of these couplings at $ep$ collisions
with high center-of-mass energy and high integrated luminosity provides an
important opportunity to test the validity of the Standard Model and the
existence of new physics beyond the Standard Model. The quartic gauge boson
couplings can contribute directly to multi-boson production at colliders.
Therefore, we examine the potential of the process $ep \rightarrow
\nu_{e}W^{+}W^{-} j$ at the Large Hadron Electron Collider and the Future
Circular Collider-hadron electron to study non-standard $WWWW$ couplings in a
model independent way by means of the effective Lagrangian approach. We present
an investigation on measuring $W^{+}W^{-}$ production in pure leptonic and
semileptonic decay channels. In addition, we calculate the sensitivity limits
at $95\%$ Confidence Level on the anomalous $\frac{f_{M0}}{\Lambda^{4}}$,
$\frac{f_{M1}}{\Lambda^{4}}$, $\frac{f_{M7}}{\Lambda^{4}}$,
$\frac{f_{S0}}{\Lambda^{4}}$, $\frac{f_{S1}}{\Lambda^{4}}$,
$\frac{f_{T0}}{\Lambda^{4}}$, $\frac{f_{T1}}{\Lambda^{4}}$ and
$\frac{f_{T2}}{\Lambda^{4}}$ couplings obtained by dimension-8 operators
through the process $ep \rightarrow \nu_{e}W^{+}W^{-} j$ for the Large Hadron
Electron Collider and the Future Circular Hadron Electron Collider's different
center-of-mass energies and integrated luminosities. Our results show that with
the process $ep \rightarrow \nu_{e}W^{+}W^{-} j$ at the Large Hadron Electron
Collider and the Future Circular Collider-hadron electron the sensitivity
estimated on the anomalous $WWWW$ couplings can be importantly strengthened.
