<p>The fully nonlinear Monge-Ampère equation poses substantial challenges in both analysis and computation even within the convex setting of current interest. Here, we investigate a Legendre spectral method for the Monge-Ampère equation on convex quadrilateral and hexahedral domains. To this end, we first develop several theoretical results on Legendre irrational orthogonal approximations, which play an essential role in extending spectral methods to quadrilateral and hexahedral domains. Based on these results, we develop the Legendre spectral scheme for the Monge-Ampère equation and derive error estimates using linearization and fixed-point arguments. A regularized Newton method is employed to enhance the stability of the iterative procedure. Numerical experiments indicate that the proposed method achieves convergence to the convex viscosity solution and confirm its accuracy and efficiency.</p>