  The study of curvature properties of homogeneous Finsler spaces with
$(\alpha, \beta)$-metrics is one of the central problems in Riemann-Finsler
geometry. In this paper, we consider homogeneous Finsler spaces with square
metric and Randers change of square metric. First, we derive the explicit
formulae for Ricci curvature of homogeneous Finsler spaces with these metrics.
Next, we find a necessary and sufficient condition under which a homogeneous
Finsler space with either of these metrics is of vanishing $S$-curvature. The
formulae for Ricci curvature of homogeneous Finsler spaces with square metric
and Randers change of square metric having vanishing $S$-curvature are
established. Finally, we prove that the aforesaid spaces having vanishing
$S$-curvature and negative Ricci curvature must be Riemannian.
