  The one-loop contributions to the trilinear neutral gauge boson couplings
$ZZV^\ast$ ($V=\gamma,Z,Z'$), parametrized in terms of one $CP$-conserving
$f_5^{V}$ and one $CP$-violating $f_4^{V}$ form factors, are calculated in
models with $CP$-violating flavor changing neutral current couplings mediated
by the $Z$ gauge boson and an extra neutral gauge boson $Z'$. Analytical
results are presented in terms of both Passarino-Veltman scalar functions and
closed form functions. Constraints on the vector and axial couplings of the $Z$
gauge boson $\left|g_{{VZ}}^{tu}\right|< 0.0096$ and
$\left|g_{{VZ}}^{tc}\right|<0.011$ are obtained from the current experimental
data on the $t\rightarrow Z q$ decays. It is found that in the case of the
$ZZ\gamma^\ast$ vertex the only non-vanishing form factor is $f_5^{\gamma}$,
which can be of the order of $10^{-3}$, whereas for the $ZZZ^\ast$ vertex both
form factors $f_5^{Z}$ and $f_4^{Z}$ are non-vanishing and can be of the order
of $10^{-6}$ and $10^{-5}$, respectively. Our estimates for $f_5^{\gamma}$ and
$f_5^{Z}$ are smaller than those predicted by the standard model, where
$f_4^{Z}$ is absent up to the one loop level. We also estimate the $ZZ{Z'}^{*}$
form factors arising from both diagonal and non-diagonal $Z'$ couplings within
a few extension models. It is found that in the diagonal case $f_{5}^{Z'}$ is
the only non-vanishing form factor and its real and imaginary parts can be of
the order of $10^{-1}-10^{-2}$ and $ 10^{-2}-10^{-3}$, respectively, with the
dominant contributions arising from the light quarks and leptons. In the
non-diagonal case $f_{5}^{Z^\prime}$ can be of the order of $10^{-4}$, whereas
$f_4^{Z'}$ can reach values as large as $10^{-7}-10^{-8}$, with the largest
contributions arising from the $Z'tq$ couplings.
