  The future muon collider can play as an ideal machine to search for new
physics at high energies. In this work, we study the search potential of the
heavy Higgs triplet in the Type II Seesaw mechanism at muon colliders with high
collision energy and high luminosity. The latest neutrino oscillation data are
taken into account for realizing the leptonic decay modes of the charged Higgs
bosons $(H^{\pm\pm},~H^{\pm})$ in the Type II Seesaw. We show the impact of
neutrino mass and mixing parameters on the purely leptonic decays. The pair
production of doubly charged Higgs $H^{++}H^{--}$ is through direct
$\mu^+\mu^-$ annihilation and vector boson fusion (VBF) processes at muon
collider. The associated production $H^{\pm\pm}H^{\mp}$ can only be induced by
VBF processes. We simulate both the purely leptonic and bosonic signal channels
of charged Higgs bosons in Type II Seesaw, together with the Standard Model
backgrounds. We show the required luminosity for the discovery of the charged
Higgses and the reachable limits on the leptonic decay branching fractions.
