I recommend that you study the paper by Duke and Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids (Inventiones, 1990).

Theorem 1 shows that if the number of representations $r(n,Q)$ exceeds $n^{1/2-1/176}$, then the representations become equidistributed on the relevant ellipsoid as $n\to\infty$. So the question is how to guarantee that $r(n,Q)$ is large. By Theorem 3 (see also the subsequent Corollary), it suffices that $n$ is primitively represented by some form in the spinor genus of $Q$. By Theorem 2, if we exclude finitely many explicitly computable square classes for $n$, the condition simplifies to: $n$ is primitively represented by some form in the genus of $Q$. The latter is equivalent to: $n$ is primitively represented by $Q$ modulo $4\det(a_{ij})$, where $(a_{ij})\in\mathrm{M}_3(\mathbb{Z})$ is the symmetric matrix such that $$Q(x_1,x_2,x_3)=\frac{1}{2}\sum_{i,j}a_{ij}x_ix_j.$$ Of course there is still the question of determining the exceptional square classes. Regarding that, see the proof of Theorem 2 and the references there. In particular, if $n$ is coprime to $\det(a_{ij})$, then the only exceptional square class is the set of squares (see Footnote 5 in arXiv:1402.1332).

I recommend that you study the paper by Duke and Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids (Inventiones, 1990).

Theorem 1 shows that if the number of representations $r(n,Q)$ exceeds $n^{1/2-1/176}$, then the representations become equidistributed on the relevant ellipsoid as $n\to\infty$. So the question is how to guarantee that $r(n,Q)$ is large. By Theorem 3 (see also the subsequent Corollary), it suffices that $n$ is primitively represented by some form in the spinor genus of $Q$. By Theorem 2, if we exclude finitely many explicitly computable square classes for $n$, the condition simplifies to: $n$ is primitively represented by some form in the genus of $Q$. The latter is equivalent to: $n$ is primitively represented by $Q$ modulo $4\det(a_{ij})$, where $(a_{ij})\in\mathrm{M}_3(\mathbb{Z})$ is the symmetric matrix such that $$Q(x_1,x_2,x_3)=\frac{1}{2}\sum_{i,j}a_{ij}x_ix_j.$$ It remains to determine the exceptional square classes. Regarding that, see the proof of Theorem 2 and the references given. In particular, if $n$ is coprime to $\det(a_{ij})$, then the only exceptional square class is the set of squares (see Footnote 5 in arXiv:1402.1332).