Boundary and Interior Points of the set: Rational Numbers

You have to specify what metric space you are working in. If you say that $\mathbb{Q}$ is a subset of the metric space $(\mathbb{R},d)$ where $d$ is the usual metric given by the usual absolute value then we get that $\operatorname{int}{\mathbb{Q}} = \emptyset$ since if we take any $x \in \mathbb{Q}$ and $r \in \mathbb{R}$, $r>0$, then any open ball (i.e. open interval) centered at $x$ will contain irrationals, and hence it will not lie entirely in $\mathbb{Q}$.

Also we have $\partial \mathbb{Q} = \mathbb{R}$ since the boundary is defined to be the closure minus the interior, and the closure if simply $\mathbb{R}$ since if $x \in \mathbb{R}$ then $x$ ``adheres'' to $\mathbb{Q}$, i.e. there will be some sequence of rational numbers tending to $x$ (for instance, a decimal expansion).

It depends on the topology we adopt. In the standard topology or $\mathbb{R}$ it is $\operatorname{int}\mathbb{Q}=\varnothing$ because there is no basic open set (open interval of the form $(a,b)$) inside $\mathbb{Q}$ and $\mathrm{cl}\mathbb{Q}=\mathbb{R}$ because every real number can be written as the limit of a sequence of rational numbers. It also follows that

$$\partial\mathbb{Q}=\mathrm{cl}\mathbb{Q}\setminus \mathrm{int}\mathbb{Q}=\mathbb{R}.$$

Comment on your edit: The definition of the interior of a set $C$ is the largest (wrt $\subseteq$) open set inside $C$, i.e., it is the union of all open sets in $C$. If (and only if) you can't find a basic open set inside $C$ whatsoever, then the interior of $C$ is empty.