Fix a bijection $f: \Bbb N \to \Bbb Q$. For convenience assume $\Bbb N = \Bbb Z_{>0}$.
It is clear that there is a bijective correspondence between infinite subsets $A \subseteq \Bbb Q$ and strictly increasing functions $g: \Bbb N \to \Bbb N$, by:
\begin{align}
A\subseteq \Bbb Q \quad&\longrightarrow\quad g(n) = \text{$n$th smallest $m$ such that $f(m) \in A$}\\
\{f(g(n)):n \in \Bbb N\} \quad&\longleftarrow\quad g:\Bbb N \to \Bbb N
\end{align}
An increasing function $g: \Bbb N \to \Bbb N$ is equivalently described by a mapping $h: \Bbb N \to \Bbb N$, as follows:
\begin{align}
g: \Bbb N \to \Bbb N \quad&\longrightarrow\quad h(1)=g(1),h(n) = g(n)-g(n-1)\\
g(n) = \sum_{k\le n} h(k) \quad&\longleftarrow\quad h:\Bbb N \to \Bbb N
\end{align}
Of these last functions there are $\aleph_0^{\aleph_0} = 2^{\aleph_0} = \mathfrak c$.