Does a connected countable metric space exist?

Fix $x_0 \in X $. Then, the continuous(!) map $$ \Phi: X \to \Bbb {R}, x \mapsto d (x,x_0) $$ has an (at most) countable, connected image.

Thus, the image is a whole (nontrivial!, if $X $ has more than one point) interval, in contradiction to being countable.

EDIT: On a related note, this even show's that every connected metric space with more than one point has at least the cardinality of the continuum.

As usual, there is a trivial example that gives you no intuition: the set containing one point is trivially a metric space (in the only possible way).

If the space contains more than one point, see this answer for a proof of uncountability.

If by countable you include finite sets: yes, there exists. Take a single point.

However, if the space has more than one point and is connected, it must be uncountable. It is a simple result from Urysohn's Lemma combined with the "Intermediate Value Theorem". Note that Urysohn's Lemma is also a triviality on metric spaces.

Since it is a result from Urysohn's Lemma, this also holds on normal topological spaces. Moreover, it holds under the weaker hypothesis of regularity instead of normality (one can use the fact that a countable space is Lindelöf, and a regular Lindelöf space is normal).

An example that this does not hold on general topological spaces can be seen by taking the trivial topology on any uncountable set. However, there exists even connected Hausdorff spaces which are countably infinite (see this).