There are plenty of topologies on a countable set for which all convergent sequences are eventually constant.

The most constructive example I know is the Arens-Fort space, given as example 26 in Steen and Seebach's *Counterexamples in Topology*.

This topology is constructed on the set $\mathbb{N}^2 \cup \{\infty\}$. Every point of $\mathbb{N}^2$ is isolated, and open neighborhoods of $\infty$ are defined to contain all but finitely many points from all but finitely many columns of $\mathbb{N} \times \mathbb{N}$ (it helps to draw a picture).

Obviously no nontrivial sequence converges to a point of $\mathbb{N}^2$. Suppose $(x_n)$ is a sequence in $\mathbb{N}^2$: we will show it does not converge to $\infty$. Either (a) it has points in only finitely many columns, so some neighborhood of $\infty$ misses the whole sequence, or (b) it has points in infinitely many columns, and we can find a infinite subsequence with one point in each column, which shows that some neighborhood of $\infty$ misses infinitely many of the $x_n$.

A little less constructively, any countable subset of the Stone-Cech compactification of the integers will have the property that every convergent sequence is eventually constant. Using this fact, you obtain $2^{2^{\aleph_0}}$ non-homeomorphic countable spaces, none of which have any nontrivial convergent sequences.