Many real-world problems are represented using Graph theory. Graphs are the models of relations. A graph is an appropriate way of representing information involving relationship between objects where the objects are represented by vertices and the relations are represented by edges. We need to design a fuzzy graph model, when there is vagueness in the description of the objects or in its relationships or in both. In this paper, we introduced Cube Fuzzy matching and Cube perfect Fuzzy matching for Fuzzy graph based on vertices. We proved the necessary condition under which they are equivalent and also proved that, for a particular condition, a perfect cube Fuzzy matching is not a (3,k) regular Fuzzy graph. We also discussed perfect Cube Fuzzy matching and Cube Fuzzy matching number for cycle graph.