This paper provides a comprehensive derivation of geodesicequations in a two-dimensional Finsler space characterized by a Matsumoto-type metric. It further exploresthe geometric applications of these equations on various surfaces, including cylinders,spheres, pseudo-spheres, and catenoids. Using illustrative examples and graphs, we analyze the geometric properties of geodesics for different parametric forms of these surfaces.The results enhance our understanding of geodesic behaviorin Finsler geometry and shedlight on curvature and geometric structures in diverse contexts.