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±μZ′. 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 Lμ-Lτ theory, into upper limits on the Z′ coupling constant at the level of 5×10-2-1 for MZ′≤6 GeV/c2. © 2020 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the "https://creativecommons.org/licenses/by/4.0/" Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP3.