Gigabit-Capable Passive Optical Network (GPON) and Long Term Evolution (LTE) are two promising broadband access technologies with high-capacity and QoS services for wired access networks and wireless access networks, respectively. A convergence of GPON and LTE networks is proposed to take the bandwidth advantage of optical networks and the mobility feature of wireless communications in this paper. First, GPON-LTE Converged Network Architecture (GLCNA) is presented and especially the concept of Optical Network Unit-evolved NodeB (ONU-eNB). Second, the QoS mapping strategy is defined for GPON-LTE converged networks. Third, a Synchronous Interleaved Dynamic Bandwidth Assignment (SIDBA) scheme is proposed to alleviate the asynchrony problem in upstream bandwidth allocation due to the cycle time of GPON (usually is between 1ms and 2ms) and the frame size of LTE (usually is 5ms or 10ms) are mismatched. The simulation results show that the proposed SIDBA scheme can effectively enhance the system performance for different polling cycle time/frame size pairs in terms of network throughput and packet delay, especially the polling cycle time/frame size pair is 2ms/5ms performs the best.