This article focuses on the theoretical and numerical analysis of the eigenvalues associated with a non-singular fractional second-order Sturm-Liouville problem. We employ a fractional-order modified operational matrix method to approximate the eigenvalues, transforming the Sturm-Liouville problem into a sparse, non-singular linear system, which is then solved using the continuation method. Theoretical findings for the problem are presented and proven, with the fractional derivative defined in the Modified Atangana-Baleanu Fractional Derivative sense. The numerical results validate the accuracy of the proposed algorithm.