We estimate the expected precision at a multi-TeV muon collider for measuring the Higgs boson couplings with electroweak gauge bosons, $HVV$ and $HHVV\ (V=W^\pm,Z)$, as well as the trilinear Higgs self-coupling $HHH$. At very high energies both single and double Higgs productions rely on the vector-boson fusion (VBF) topology. The outgoing remnant particles have a strong tendency to stay in the very forward region, leading to the configuration of the ``inclusive process'' and making it difficult to isolate $ZZ$ fusion events from the $WW$ fusion. In the single Higgs channel, we perform a maximum likelihood analysis on $HWW$ and $HZZ$ couplings using two categories: the inclusive Higgs production and the 1-muon exclusive signal. In the double Higgs channel, we consider the inclusive production and study the interplay of the trilinear $HHH$ and the quartic $VVHH$ couplings, by utilizing kinematic information in the invariant mass spectrum.