Which of the following is the accepted mathematical practice?
Any segment $(a, b)$ or interval $[a, b]$ contains only real numbers. If you want all the rational numbers between $a$ and $b$, you have to write $\{a < q < b | q \in \mathbb{Q}\}$ explicitly; saying "$(a, b)$ in $\mathbb{Q}$" is nonsense.
Any segment $(a, b)$ or interval $[a, b]$ contains every number between $a$ and $b$ (including $a$ and $b$ in the interval case). By convention, if nothing to the contrary is specified, then "every number" means every real number. But if you see "$(a, b)$ in $\mathbb{Q}$", you should read $\{a < q < b | q \in \mathbb{Q}\}$.