Based on the separation-of-variables method and Mathieu function theory, we rigorously solved the Schrödinger equation for a two-dimensional infinite-depth elliptical quantum well, obtaining the energy level distributions and corresponding wave functions for various eccentricities e_r and quantum states. By separating variables in elliptical coordinates, the wave function is expressed as the product of radial Mathieu functions and angular Mathieu functions. Combining the angular periodicity boundary condition with the infinite-well boundary condition, we derived the characteristic equation that determines the discrete energy eigenvalues. Through analysis of the zero-distribution patterns of Mathieu functions under different parities and angular quantum numbers m, the energy spectra were numerically computed, and the dependence of the energy levels on the eccentricity was investigated in detail. Using the analytical expressions for the wave functions, the properties of the wave functions in different quantum states were thoroughly analyzed.