Elliptic-curve group cryptography (ECC) has been widely recognized as a robust and efficient cryptographic system in the classical era. However, with the emergence of quantum computing, traditional public key cryptosystems are at risk of being broken, which has prompted the exploration of alternative cryptographic methods. This research paper explores the fundamental aspects of elliptic curve groups in cryptography, their properties, and their application in cryptographic protocols in quantum era, focusing on the potential vulnerabilities of ECC in the quantum computing age and the efforts to develop post-quantum secure ECC-based schemes.