Just out of curiosity is the following a folner sequence of $\mathbb{Q}$
$$\left\{\frac{2m+1}{2n}:m\in\mathbb{Z},n\in\mathbb{Z}_{\neq 0},n<d\right\}\cup\left\{\frac{j}{2k+1}:j,k\in\mathbb{Z},k<d^d\right\}$$
Folner sequences of $A=\mathbb{Q}$ are defined by the condition
$$\lim_{d\to\infty}\frac{|gA_d\Delta A_d|}{|A_d|}=0$$
Where for group $G$ that acts on $A$, $g\in G$ and $\left\{A_d\right\}_{d\in\mathbb{N}}$ are a sequence of finite subsets of $A$.