This paper explores the application of Quadratic Unconstrained Binary Optimization (QUBO) models in solving the Travelling Salesman Problem (TSP) through Quantum Annealing algorithms and Graph Neural Networks. Quantum Annealing (QA), a quantum-inspired optimization method that exploits quantum tunneling to escape local minima, is used to solve QUBO formulations of TSP instances on Coherent Ising Machines (CIMs). The paper also presents a novel approach where QUBO is employed as a loss function within a GNN architecture tailored for solving TSP efficiently. By leveraging GNN's capability to learn graph representations, this method finds approximate solutions to TSP with improved computational time compared to traditional exact solvers. The paper details how to construct a QUBO model for TSP by encoding city visits into binary variables and formulating constraints that guarantee valid tours. It further discusses the implementation of QUBO-based Quantum Annealing algorithm for TSP (QQA-TSP) and its feasibility demonstration using quantum simulation platforms. In addition, it introduces a Graph Neural Network solution for TSP (QGNN-TSP), which learns the underlying structure of the problem and produces competitive solutions via gradient descent over a QUBO-based loss function. The experimental results compare the performance of QQA-TSP against state-of-the-art classical solvers such as dynamic programming, Concorde, and Gurobi, while also presenting empirical outcomes from training and evaluating QGNN-TSP on various TSP datasets. The study highlights the promise of combining deep learning techniques with quantum-inspired optimization methods for solving NP-hard problems like TSP, suggesting future directions for enhancing GNN architectures and applying QUBO frameworks to more complex combinatorial optimization tasks.