The Internet of Things (IoT) is expanding rapidly, which has created a need for sophisticated computational frameworks that can handle the data and security requirements inherent in modern IoT applications. However, traditional cloud computing frameworks have struggled with latency, scalability, and security vulnerabilities. Quantum-Edge Cloud Computing (QECC) is a new paradigm that effectively addresses these challenges by combining the computational power of quantum computing, the low-latency benefits of edge computing, and the scalable resources of cloud computing. This study has been conducted based on a published literature review, performance improvements, and metrics data from Bangladesh on smart city infrastructure, healthcare monitoring, and the industrial IoT sector. We have discussed the integration of quantum cryptography to enhance data integrity, the role of edge computing in reducing response times, and how cloud computing's resource abundance can support large IoT networks. We examine case studies, such as the use of quantum sensors in self-driving vehicles, to illustrate the real-world impact of QECC. Furthermore, the paper identifies future research directions, including developing quantum-resistant encryption and optimizing quantum algorithms for edge computing. The convergence of these technologies in QECC promises to overcome the existing limitations of IoT frameworks and set a new standard for the future of IoT applications.