In this dissertation, I present predictions for the scattering cross sections of Higgs boson pair and triple production via photon fusion at future muon colliders within the framework of the Higgs Triplet Model, a beyond-the-Standard Model theory that generates neutrino masses. High-energy muon beams can emit collinear photons, leading to the production of multi-Higgs final states through loop-induced processes. Thus, I focus on the processes and via photon fusion, utilizing the Effective Photon Approximation for cross-section predictions. Our analysis includes contributions from singly and doubly charged Higgs bosons and their couplings to CP-even scalars. I have established my own parameter scanning method, as taking the masses of the scalars as inputs is challenging due to model consistency constraints. I compare total cross sections and kinematic distributions with Standard Model predictions, utilizing automated one-loop calculators for numerical amplitude evaluations and Monte Carlo phase-space integrators for collider simulations at center-of-mass energies ranging from 2 TeV to 10 TeV.