Given an open set $U \subset \mathbb{R}$, is it possible to express $U$ as a disjoint union of $(a,b)$'s where $a,b \in Q$ for some dense set $Q$?

Question

I am aware that any open subset $U \in \mathbb{R}$ may be expressed as a countable disjoint union of open intervals.

Now, if we are given a dense subset $Q \subset \mathbb{R}$, is it possible to express \begin{equation} U=\cup_{i=1}^\infty (a_i,b_i) \end{equation} where $a_i,b_i \in Q$ for all $i \in \mathbb{N}$ and the union $\cup_{i=1}^\infty$ is disjoint.

I guess this is true, but not sure how to verify rigorously.. Could anyone please help me?