Using data collected by the Belle II experiment during 2018 at the SuperKEKB collider, we perform the first searches for the invisible decay of a \(Z^{\prime}\) in the process \(e^+ e^- \to \mu^+ \mu^- Z^{\prime}\) and of a lepton-flavor-violating \(Z^{\prime}\) in \(e^+ e^- \to e^{\pm} \mu^{\mp} Z^{\prime}\). We do not find any excess of events and set 90\(\%\) credibility level upper limits on the cross sections of these processes. We translate the former, in the framework of an \(L_{\mu}-L_{\tau}\) theory, into upper limits on the \(Z^{\prime}\) coupling constant at the level of \(5 \times 10^{-2}\div1\) for \(M_{Z^\prime}\leq 6\) GeV/\(c^2\).