We study in the context of the $U(1)_{B-L}$ model the production and decays of the heavy Higgs boson $H$ via the association between the $Z'$ boson and the new heavy Higgs boson $H$. We consider the $\mu^+\mu^- \to (Z, Z') \to ZH \to l^\pm l^\mp W^\pm W^\mp$ signal with the subsequent decay of $Z$ to pairs of $l^\pm l^\mp$ with $l=e, \mu$ and $H$ to pairs of $W^\pm W^\mp$. The projections of new physics at the muon collider with the benchmark center-of-mass energies of $\sqrt {s}=3, 4, 5, 6, 7$ TeV and conservative integrated luminosities of ${\cal L}_{\rm con}=1, 2, 3, 4, 10$ ${\rm ab^{-1}}$, is of the order of 684 expected events. This scenario demonstrates the importance of measuring the heavy Higgs boson $H$ at the future muon collider. In another scenario, in addition to the $\mu^+\mu^- \to (Z, Z') \to ZH \to l^\pm l^\mp W^\pm W^\mp$ signal, we consider Vector Boson Scattering (VBS) rates of $\mu^+\mu^- \to WWZ$, and the Leptonic, Semi-leptonic, and Hadronic channels of the $W^\pm$ boson for the signal. In this case, the number of events expected to be produced at the future muon collider is $N=329\,551$, which is a promising scenario.