In this paper, sufficient conditions for the existence of Zeno behavior in a class of hybrid systems are given; these are the first sufficient conditions on Zeno of which the authors are aware for hybrid systems with nontrivial dynamics. This is achieved by considering a class of hybrid systems termed diagonal first quadrant (DFQ) hybrid systems. When the underlying graph of a DFQ hybrid system has a cycle, we can construct an in finite execution for this system when the vector fields on each domain satisfy certain assumptions. To this execution, we can associate a single discrete time dynamical system that describes its continuous evolution. Therefore, we reduce the study of executions of DFQ hybrid systems to the study of a single discrete time dynamical system. We obtain sufficient conditions for the existence of Zeno by determining when this discrete time dynamical system is exponentially stable.