A search for a narrow Z$'$ gauge boson with a mass between 5 and 70 GeV resulting from an $L_{\mu}$ $-$ $L_{\tau}$ $U(1)$ local gauge symmetry is reported. Theories that predict such a particle have been proposed as an explanation of various experimental discrepancies, including the lack of a dark matter signal in direct-detection experiments, tension in the measurement of the anomalous magnetic moment of the muon, and reports of possible lepton flavor universality violation in B meson decays. A data sample of proton-proton collisions at a center-of-mass energy of 13 TeV is used, corresponding to an integrated luminosity of 77.3 fb$^{-1}$ recorded in 2016 and 2017 by the CMS detector at the LHC. Events containing four muons with an invariant mass near the standard model Z boson mass are analyzed, and the selection is further optimized to be sensitive to the events that may contain Z $\to$ Z$'\mu\mu$ $\to$ 4$\mu$ decays. The event yields are consistent with the standard model predictions. Upper limits of 10$^{-8}$-10$^{-7}$ at 95% confidence level are set on the product of branching fractions $\mathcal{B}$(Z $\to$ Z$'\mu\mu$) $\mathcal{B}$(Z$'$ $\to$ $\mu\mu$), depending on the Z$'$ mass, which excludes a Z$'$ boson coupling strength to muons above 0.004-0.3. These are the first dedicated limits on $L_{\mu}$ $-$ $L_{\tau}$ models at the LHC and result in a significant increase in the excluded model parameter space. The results of this search may also be used to constrain the coupling strength of any light Z$'$ gauge boson to muons.