The third generation partnership project (3GPP) has addressed the feasibility of interworking and specified the interworking architecture and security architecture for third generation (3G)-wireless local area network (WLAN), it is developing, system architecture evolution (SAE)/ long term evolution (LTE) architecture, for the next generation mobile communication system. To provide a secure 3G-WLAN interworking in the SAE/LTE architecture, Extensible authentication protocol-authentication and key agreement (EAP-AKA) is used. However, EAP-AKA have several vulnerabilities. Therefore, this paper not only analyses the threats and attacks in 3G-WLAN interworking but also proposes a new authentication and key agreement protocol based on EAP-AKA. The proposed protocol combines elliptic curve Diffie-Hellman (ECDH) with symmetric key cryptosystem to overcome the vulnerabilities. The proposed protocol is used in hybrid coupled 3G-WLAN convergence network to analyse its efficiency in terms of QoS metrics, the results obtained using OPNET 14.5 shows that the proposed protocol outperforms existing interworking protocols both in security and QoS.