Future 6G networks must handle ever-increasing demands for low-latency, high-throughput, and complex multiservice requirements. In this paper, we propose a hybrid quantum-classical approach that combines Software-Defined Networking (SDN) with the Quantum Approximate Optimization Algorithm (QAOA) to tackle routing optimization. By modeling routing as a Quadratic Unconstrained Binary Optimization (QUBO) problem and leveraging a gate-based quantum computer, our framework demonstrates the feasibility of using variational quantum methods for network tasks. We implement proof-of-concept scenarios in IBM Qiskit, outlining how a microservices-based SDN controller can incorporate a quantum solver for route calculation. Results highlight that QAOA can identify low-latency paths and accommodate flow constraints, pointing to the significant future promise of quantum SDN solutions as quantum hardware evolves. We also discuss the potential for distributed quantum controllers, deployment considerations, and strategies to mitigate noise in current NISQ devices. Our findings lay foundations for deeper exploration into quantum-based network optimization under practical 6G conditions.