Theories beyond the standard model often predict the existence of an additional neutral boson, the Z^{'}. 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^{'} in the process e^{+}e^{-}→μ^{+}μ^{-}Z^{'} and of a lepton-flavor-violating Z^{'} in e^{+}e^{-}→e^{±}μ^{∓}Z^{'}. 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_{μ}-L_{τ} theory, into upper limits on the Z^{'} coupling constant at the level of 5×10^{-2}-1 for M_{Z^{'}}≤6 GeV/c^{2}.