  Non-Abelian structure of the Standard Model predicts the self-interactions of
gauge bosons triple gauge couplings (TGC) and quartic gauge couplings (QGC). On
the other hand, it is also important to determine the deviations from Standard
Model (SM) via anomalous triple gauge couplings (aTGC) and anomalous quartic
gauge couplings (aQGC) to test the nature of the Standard Model (SM) and to see
the effects of new physics arising from the beyond standard model (BSM). In
this study, we focus on the $e^-p \to j Z\gamma \nu_e \to j (l^{+} l^{-})\gamma
\nu_e$ process to examine the $WWZ\gamma$ anomalous quartic gauge couplings
(aQGC) at the Future Circular Collider-hadron electron (FCC-he) with
center-of-mass energy $\sqrt{s}=5.29$ TeV using with a model-independent way in
the effective theory approach. A Cut-based method are applied to analyse the
signal and relevant SM background. Using with the total cross-sections, we
constrained on the anomalous $ f_ {T,i=0,1,2,5,6,7}/\Lambda^4$ couplings
arising from dimension-8 operators at $95\%$ Confidence Level (C.L.) for the
dilepton decay of the $Z$-boson. The results are composed with integrated
luminosities of ${\cal L}=1,2$ and $3$ $\rm ab^{-1}$ under the systematic
uncertainties of $\%0$, $\%5$ and $\%10$. In the calculations, we use the form
factor to eleminate the effects of the violation of unitarity. With this, we
obtained the limits for the energy scale of the form factor $\Lambda=1$ TeV. In
addition to all this, we use the adventages of $ep$ collisions for analysing
the process with much cleaner environment and lower background effects
comparing with the $pp$ collisions. The obtained sensitivities on $ f_
{T,i=0,1,2,5,6,7}/\Lambda^4$ are comparable with the experimental results and
related phenomenological studies in the literature.
