The muon collider is one of the most attractive projects for the next generation colliders, capable of providing leptonic collisions at the energies of several TeV. The first study about the muon collider prospects for the measurement of the Higgs boson production cross section 𝜎𝐻 times the branching ratio for the Higgs boson decay to a pair of muons 𝐵𝑅(𝐻→𝜇𝜇) is presented. The analysis is performed considering a 3-TeV muon collider and assuming an integrated luminosity of 1 ab⁻¹. This study is performed with a full simulation of the muon collider detector. It is shown that the expected number of signal events in the di-muon system invariant mass range from 105 to 145 GeV is 25.8 ± 9.9. The sensitivity on the measurement of 𝜎𝐻×𝐵𝑅(𝐻→𝜇𝜇) is estimated to be 38%.