We present a quantum spectral method for non-periodic partial differential equations (PDEs) with arbitrary boundary conditions, achieving polylogarithmic complexity O((log N)^c), where N is the number of grid points, compared to the polynomial complexity O(N^c) of classical approaches. Our work extends the quantum spectral approach of Liu et al. (2024) beyond periodic problems, enabling the quantum solution of a broader class of PDEs from engineering. To enforce homogeneous Dirichlet boundary conditions, we extend the problem domain to twice its original size using an antisymmetric reflection of all field variables. The PDE is discretised with spectral Fourier basis functions on the extended domain and solved via the quantum Fourier transform (QFT). For non-homogeneous Dirichlet boundary conditions, we decompose the solution into a prescribed component that satisfies the non-homogeneous boundary conditions and an unknown component determined by solving the PDE. We verify the accuracy and algorithmic scaling of the proposed approach through numerical experiments in one and two dimensions, solving Poisson problems with arbitrary Dirichlet boundary conditions, high-order PDEs and stochastic PDEs. Our experiments confirm the optimal convergence of the quantum solution and the polylogarithmic increase of the number of quantum gates with increasing problem size.