  In this paper, we study the moments of the Margulis $\alpha$-function
integrating over expanding translates of a unipotent orbit in
$\operatorname{SL}_3(\mathbb{R})/\operatorname{SL}_3(\mathbb{Z})$. We show that
for some $\lambda>1$ the $\lambda$-moments of the Margulis $\alpha$-function
over expanding translates of a unipotent orbit are uniformly bounded, under
suitable Diophantine conditions of the initial unipotent orbit.
  As an application, we prove that for any indefinite irrational ternary
quadratic form $Q$ with suitable Diophantine conditions and $a<b$ the number of
integral vectors of norm at most $T$ satisfying $a<Q(v)<b$ is asymptotically
equivalent to $\mathsf{C}_Q(b-a)T$ as $T$ tends to infinity, where the constant
$\mathsf{C}_Q>0$ depends only on $Q$.
