A solvmanifold is a manifold $ M $ which admits a transitive action by a solvable Lie group. A Riemannian homogeneous space is a Riemannian manifold whose isometry group acts transitively.
Let $ M $ be a compact solvmanifold which is Riemannian homogeneous with respect to some metric $ g $ . Can we conclude that $ M $ is a torus?
Yes. A Riemannian homogeneous manifold has fundamental group with finite commutator subgroup.
Transitive action by compact Lie group implies almost abelian fundamental group
But the fundamental group of a solvmanifold is torsion free so a finite subgroup is trivial. Thus a Riemannian homogeneous solvmanifold has abelian fundamental group. Since the fundamental group of a compact solvmanifold is always torsion free and finitely generated then we can conclude the fundamental group is $ \mathbb{Z}^n $. But it is a theorem of Auslander that a compact solvmanifold is determined up to diffeomorphism by it's fundamental group. So a compact Riemannian homogeneous solvmanifold must be a torus.
Indeed this result generalizes from solvmanifolds to all aspherical manifolds that are compact Riemannian homogeneous see
Riemannian homogeneous aspherical iff flat torus
Also note that the linked question proves something much stronger: a Riemannian homogeneous aspherical closed manifold is not just diffeomorphic to a torus but in fact the original homogeneous Riemannian metric must have been flat to begin with.