  In the post-LHC era, the muon collider represents a frontier project capable
of providing high-energy and high-luminosity leptonic collisions among future
lepton-lepton particle accelerators. In addition, it provides significantly
cleaner final states than those produced in hadron collisions. With this
expectation in mind, in this article, we research the sensitivity of the
anomalous neutral gauge boson couplings $Z\gamma\gamma\gamma$, $ZZ\gamma
\gamma$, and $ZZZ\gamma$ defined by dimension-8 operators, through the
$\mu^+\mu^- \to \mu^+\mu^-Z\gamma$ signal, with the $Z$-boson decaying to
neutrino pair. The projections of new physics at the future muon collider with
the center-of-mass energy of $\sqrt{s}=10$ TeV, integrated luminosity of ${\cal
L}=10$ $\rm ab^{-1}$, and systematic uncertainties of $\delta_{sys}=0\%, 3\%,
5\%$, for extraction of expected sensitivity on the anomalous $f_
{T,j}/\Lambda^4$ couplings at $95\%$ confidence level, are of the order of
${\cal O}(10^{-4}- 10^{-3})$. Compared with the research of the ATLAS and CMS
Collaborations on the anomalous quartic gauge couplings, we find that the
high-luminosity future muon collider could have better sensitivity.
