In this work, a quantum Taylor series expansion and Least squares-based Fractional-step Lattice Boltzmann Method (TLFS-LBM) is proposed to simulate incompressible flows on nonuniform meshes. Most existing quantum LBMs face two critical challenges: with the relaxation time s fixed at 1, a fixed mesh resolution corresponds to only a single Reynolds number, and the streaming step, typically implemented through the quantum walk method, is limited to uniform meshes. To address these issues, we retain τ=1 and decompose the quantum TLFS-LBM into predictor and corrector steps using a fractional-step method. The predictor step is implemented on a quantum circuit, but the resulting viscosity generally differs from the physical viscosity. A diffusion equation is therefore solved classically in the corrector step to restore the correct physical viscosity. Moreover, the streaming step is expressed in the form of A·f^eq using Taylor series expansion and least-squares approximation, with singular value decomposition applied to factorize matrix A into unitary matrices for quantum implementation. Since A depends solely on mesh point coordinates, the proposed method is applicable to arbitrary mesh types. Validation through two-dimensional incompressible isothermal and thermal flow simulations confirms that the quantum TLFS-LBM achieves accuracy comparable to its classical counterpart, while extending the applicability of quantum LBM to non-uniform meshes and curved boundaries.