I am trying to assemble the answers to the question(s) that were hashed out in the comments (and also in a separate discussion with Jonny Evans). This answer is community wiki since it is the outcome of collaborative discussion. Please feel free to edit this.
Rephrasing of question: does there exist a non-constant holomorphic curve in any symplectic manifold? (presumably, for generic choice of compatible J) Can we say more in dimension 4?
- A generic K3 surface (which is a symplectic 4-manifold) does not have any curves, so the answer to the question in complete generality is "no". We therefore reinterpret the question to be about finding a large class of 4-manifolds for which we can say something.
- If we drop the non-constant condition, there are the trivial (constant) holomorphic curves. This is why we require non-constant holomorphic curves.
- If we allow ourselves to find a $J$-holomorphic curve for a very special (not generic!) almost complex structure $J$, it suffices to find an embedded symplectic surface and then construct $J$ to make this surface $J$-holomorphic. In dimension 4, we can find a symplectic surface by finding a Donaldson divisor.
- If there exists a $J$-holomorphic sphere in $N$, then there exist $J$-holomorphic maps from domains of all genus, by composing with a branched cover.
- There are two obvious infinite families of examples for which we can find non-constant holomorphic curves. The first are products of symplectic manifolds with surfaces. The second family of examples is obtained by blowing up a symplectic 4-manifold.
- Another family of examples come from 4-manifolds $(N, \omega)$ for which the Gromov-Taubes invariant is non-vanishing. For instance, if $c_1(TN) \ne 0$.