The conjecture follows from Theorem 2.1 in Hsia-Jöchner: Almost strong approximations for definite quadratic spaces, Inventiones 129 (1997), 471-487. The paper is available here.
The details of this implication are nontrivial. Let $q=\ell^s$ be a prime power (so $\ell$ is the underlying prime), and let $B\in\mathcal{B}$. Let $k$ be large and divisible by $\varphi(q)$. Then $2^k$ is large, and $Q(B)\equiv 1\equiv 2^k\pmod{q}$. By a standard lifting argument, there exists a primitive $v_\ell\in B+q\mathbb{Z}_\ell^4$ such that $Q(v_\ell)=2^k$. Let us also define $v_2\in\mathbb{Z}_2^4$ and $v_{13}\in\mathbb{Z}_{13}^4$ as the integer vector $(2^{k/2},0,0,0)$. Note that $v_{13}$ is primitive, while $v_2$ is not. Note also that $Q(v_2)=Q(v_{13})=2^k$. Now we apply the above mentioned theorem for $T:=\{2,13,\ell\}$ and $\alpha:=2^k$ and $\{v_2,v_{13},v_\ell\}$. The conditions of the theorem are satisfied, because $Q$ is isotropic over $\mathbb{Q}_2$ (cf. Section IV.2 in Serre: A course in arithmetic). We conclude that there exists $A\in\mathbb{Z}^4$ such that $Q(A)=2^k$ and $A\equiv v_p\pmod{p^s}$ for $p\in T$. In particular, $A\in\mathcal{A}$ and $A\equiv B\pmod{q}$. We are done.