It is well known that every continuous injective map $\mathbb{R}\rightarrow\mathbb{R}$ is monotone. This statement is false for maps $\mathbb{Q}\rightarrow\mathbb{Q}$. (That is becaus $\mathbb{Q}$ is not complete. You can change from increasing to decreasing and vice versa in each irrational "hole").
Is it true that every homeomorphism of $\mathbb{Q}$ is monotone?
No, homeomorphisms of $\mathbb{Q}$ need not be monotone. For an irrational $c > 0$, let
$$h_c(x) = \begin{cases}x &, \lvert x\rvert < c\\ -x &, \lvert x\rvert > c. \end{cases}$$
Then $h_c$ is a non-monotonic homeomorphism of $\mathbb{Q}$.