Let $F$ be a subfield of $\mathbb{Q}$. Then $F$ has the identity $1$, so $F$ contains $\mathbb{Z}$. Since $F$ is a field, for any nonzero $n\in\mathbb{Z}$, there is the inverse $\frac{1}{n}$ in $\mathbb{Q}$. Thus for every $\frac{p}{q}\in \mathbb{Q}$, where $p\in\mathbb{\mathbb{Z}}$ and $q\in\mathbb{Q}\setminus{0}$, $\frac{p}{q}=p\cdot \frac{1}{q}\in F$ and $F=\mathbb{Q}$.