A cone has one edge. The edge appears at the intersection of of the circular plane surface with the curved surface originating from the cone's vertex.
This question could be confusing since it depends on how one defines "edge" and "plane." If it is considered that the cone has an infinite amount of intersecting planes, there are an infinite number of edges where each of these planes intersect with each other. If one considers the term "edge" to mean the boundary of a geometric plane figure, then a cone has one edge along the perimeter of its flat base.