The finite element method (FEM) is central to computational mechanics, yet large-scale simulations---particularly in high spatial dimensions or in problems requiring extremely fine resolution---can exceed the capabilities of classical high-performance computing. Quantum algorithms for partial differential equations (PDEs) have shown that certain exponentially large linear systems can be treated with polylogarithmic dependence on the system size, but existing approaches are largely restricted to structured finite-difference or spectral discretizations on simple geometries. In this contribution, we present Qu-FEM, a fault-tolerant quantum algorithmic framework for the finite element method for elliptic PDEs. Our approach constructs block-encodings of global FEM stiffness and mass matrices by emulating the classical assembly process on a quantum computer. We introduce two quantum primitives: (i) the unit of interaction, which aggregates all couplings between a fixed pair of local nodes across the mesh using a constant-size linear combination of unitaries, and (ii) the local-to-global node number indicator matrix, which provides a reversible encoding of connectivity for general assembly tasks. For a modified Poisson equation on Cartesian meshes with tensor-product Lagrange elements of order $p$ in $d$ dimensions and $2^n$ grid points per direction, we obtain explicit Clifford+T circuits whose block-encodings of global arrays require only $\tilde{O}(d^2 p^2 n)$ gates. For spatially varying coefficients, we show how to perform Gauss-Legendre numerical integration directly on the quantum computer to assemble both global matrices and load vectors within the same framework. Dirichlet boundary conditions are enforced via a quantum implementation of the method of Lagrange multipliers, while Neumann data are incorporated through assembled boundary force vectors. Ultimately, this formulation offers a concrete route toward leveraging quantum architectures for large-scale computational mechanics.