As Emil pointed out in a comment, the answer to this question will depend on what else is available in your intuitionistic logic. If that background logic is strong enough, then (Q) will imply the law of the excluded middle. Specifically, suppose we're working in a higher-order intuitionistic logic (also called intuitionistic type theory) such as arises as the internal logic of topoi, and suppose we adjoin (Q) as a general principle, i.e., with arbitrary formulas $\phi$ and with the variable $x$ ranging over an arbitrary type. Then we get classical logic, in the form $(\neg\neg\alpha)\to\alpha$ (which is known to imply the law of the excluded middle) as follows.

Given $\alpha$, let $A=\{x\in1:\alpha\}$. That is, $A$ is the subset of the singleton 1 that contains the unique element of 1 iff $\alpha$ is true. Then, for any formula $\phi$, we have that $(\exists x\in A)\,\phi$ is equivalent to $\alpha\land\phi$ and that $(\forall x\in A)\,\phi$ is equivalent to $\alpha\to\phi$. In particular, if we apply (Q) with $\bot$ (the always false proposition) in the role of $\phi$ and with the quantifiers ranging over $A$, then the left side of (Q) is equivalent to $\neg(\alpha\to\bot)$ and thus to $\neg\neg\alpha$, while the right side of $Q$ becomes equivalent to $\alpha\land\neg\bot$ and thus to $\alpha$. So this particular instantiation of (Q) says $(\neg\neg\alpha)\to\alpha$.