  In the paper, the inclusive production of heavy quarkonium $\eta_Q$ ($Q=b$ or
$c$) via $Z$ boson decays within the framework of nonrelativistic QCD (NRQCD)
effective theory are studied. The contributions from the leading color-singlet
and color-octet Fock states are considered. Total and differential decay widths
for the inclusive decays $Z \to \eta_c+X$ and $Z \to \eta_b+X$ are presented.
It is found that the decay $Z\to \eta_Q +X$ is dominated by the $^3S_1^{[8]}$
component, so the decays can be inversely adopted to determine the values of
the long-distance matrix elements $\langle {\cal
O}^{\eta_{c}}(^{3}S_{1}^{[8]})\rangle$ and $\langle {\cal
O}^{\eta_{b}}(^{3}S_{1}^{[8]})\rangle$ respectively. The numerical results show
that at an $e^+e^-$ collider running at the $Z$ pole with a high luminosity
around $10^{35}{\rm cm}^{-2}{\rm s}^{-1}$ (a super $Z$ factory), there are
about $4.5\times 10^7$ $\eta_c$ meson events and $6.2\times 10^5$ $\eta_b$
meson events to be produced per operation year, and the inclusive decays may be
used for clarifying some problems on the heavy quarkonium $\eta_Q$ and NRQCD.
