This paper presents a search for a new Z′ vector gauge boson with the ATLAS experiment at the Large Hadron Collider using pp collision data collected at $ \sqrt{s} $ = 13 TeV, corresponding to an integrated luminosity of 139 fb$^{−1}$. The new gauge boson Z′ is predicted by L$_{μ}$ − L$_{τ}$ models to address observed phenomena that can not be explained by the Standard Model. The search examines the four-muon (4μ) final state, using a deep learning neural network classifier to separate the Z′ signal from the Standard Model background events. The di-muon invariant masses in the 4μ events are used to extract the Z′ resonance signature. No significant excess of events is observed over the predicted background. Upper limits at a 95% confidence level on the Z′ production cross-section times the decay branching fraction of pp → Z′μμ → 4μ are set from 0.31 to 4.3 fb for the Z′ mass ranging from 5 to 81 GeV. The corresponding common coupling strengths, g$_{Z′}$, of the Z′ boson to the second and third generation leptons above 0.003 – 0.2 have been excluded.