Is there a bijection from a bounded open interval of $\mathbb{Q}$ onto $\mathbb{Q}$? It is easy to create a bijection between two bounded open intervals of $\mathbb{R}$, such as:
$$
\begin{align}
 f : (a,b) &\to (\alpha,\beta) \\
 x &\mapsto \alpha+(x-a)(\beta-\alpha).
\end{align}
$$
It is also possible to biject a bounded open interval of $\mathbb{R}$ onto the whole of $\mathbb{R}$, e.g.:
$$
\begin{align}
 f : (a,b) &\to \mathbb{R} \\
 x &\mapsto \tanh^{-1} x.
\end{align}
$$
Consider now the set of rational numbers $\mathbb{Q}$. The bijection between two bounded open intervals still holds, but is it possible to biject:
$$
f : (p,q) \to \mathbb{Q}
$$
where $p,q\in\mathbb{Q}$ and $(p,q) = \{x\in \mathbb{Q} : p < x < q\}$?

Cantor showed that any two countable densely ordered sets with no first or last element are order isomorphic. One can build the isomorphism using his Back and Forth Method.
If all we want is a bijection, we can more simply note that both sets are countably infinite. 
