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 a large positive integer divisible by $\varphi(q)$. Then the $2$-power $\alpha:=2^k$ is large, and $Q(B)\equiv 1\equiv\alpha\pmod{q}$. By a standard lifting argument, there exists a primitive vector $v_\ell\in B+q\mathbb{Z}_\ell^4$ such that $Q(v_\ell)=\alpha$. Let us also define $v_2\in\mathbb{Z}_2^4$ and $v_{13}\in\mathbb{Z}_{13}^4$ as the integer vector $(2^{k/2},0,0,0)\in\mathbb{Z}^4$. We observe that $Q(v_2)=Q(v_{13})=\alpha$, and $v_{13}\in\mathbb{Z}_{13}^4$ is primitive, but $v_2\in\mathbb{Z}_2^4$ is not. Now we apply the mentioned theorem of Hsia-Jöchner for the primes $T:=\{2,13,\ell\}$ and the above data. The conditions of the theorem are satisfied, because $Q$ is positive definite and 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 $Q(A)=\alpha$ and $A\equiv v_p\pmod{p^s}$ for all $p\in T$. In particular, $A\in\mathcal{A}$ and $A\equiv B\pmod{q}$. We are done.