Topology optimization is a powerful design method in engineering for determining the optimal material distribution within a given domain. However, the associated optimization problem is generally non-convex and large-scale. While conventional gradient methods are computationally efficient, they often get trapped in local optima.Conversely, global search methods are computationally expensive, making it difficult to obtain truly optimal solutions for real-scale problems. Quantum Hamiltonian Descent (QHD) offers a promising approach to resolve this trade-off by combining the efficiency of gradient methods with the global search capability derived from quantum tunneling. In this study, we propose a quantum algorithm using QHD to solve topology optimization problems on a Fault-Tolerant Quantum Computer (FTQC). Specifically, we encode the compliance minimization problem for truss structures into the potential term of a quantum Hamiltonian, and decompose its time evolution operator into a sequence of quantum gates using Suzuki-Trotter decomposition. Furthermore, we analyzed the computational complexity of the proposed method. Numerical experiments on a simple two-dimensional truss structure that the optimal solution can be obtained under appropriate hyperparameter settings, validating the effectiveness of the proposed approach.