The general theory of quaternionic modular forms has been studied by several authors. Especially, A. Krieg has shown that many results of the classical Siegel modular forms can be generalized to the quaternionic case. For example, Hecke's theory, the notion of Maass space, the Koecher-Maasss Dirichlet series, and so on. In this paper, the authors give a dimension formula for the space of cusp forms defined on the quaternionic half-space of degree two by using the Selberg trace formula. Moreover, they calculate the contributions of some conjugacy classes.