This paper investigates quantum computing approaches for solving the Graph Colouring Problem (GCP), a fundamental combinatorial optimization challenge with applications in scheduling and network design. Two methods are compared: a Constraint Satisfaction Problem (CSP) formulation and a Superposition-based approach, both implemented using quantum simulation libraries and virtual circuits. The graph colouring constraints are encoded into binary quadratic models (BQMs) to enable quantum optimization. Computational experiments evaluate each method’s performance in terms of solution accuracy, scalability, and execution time. Results show that while the CSP method achieves high accuracy, it faces challenges in scaling to complex graphs. In contrast, the Superposition method offers faster execution times and maintains reliable success rates across various graph instances. These findings demonstrate the potential of quantum simulation techniques to enhance the efficiency and scalability of graph colouring algorithms, providing insights into the development of future quantum–classical hybrid solutions for combinatorial optimization problems.