The homogeneous minimal hypersurfaces in $S^n$ have $g = 1,2,3,4$, or $6$ distinct (constant) principal curvatures. While the Morse index and nullity have been calculated for all such hypersurfaces having $g = 1,2,3$, it has remained an open problem to compute these quantities for any of those with $g = 4$ or $6$. In this paper, we calculate the Morse index and nullity of two homogeneous minimal hypersurfaces in $S^n$ with $g = 4$. Moreover, we observe that their Laplace spectra contain irrational eigenvalues that are not expressible in radicals.