The brief answer to your question is 'no':  For example, take $N=M$ and $J_N=J_M$.  Then the identity map of $N$ is a nonconstant pseudo-holomorphic map.

What *is* true is that the nonvanishing of the Nijnhuis tensors of the two manifolds puts nontrivial conditions (beyond merely being complex linear) on the induced map on the tangent bundles.  Depending on the algebra of the two Nijnhuis tensors when the dimension of $N$ is greater than $2$, it can indeed happen that these conditions imply that any pseudo-holomorphic mapping from $N$ to $M$ must be constant.

For example, if $N=S^6$ and $J_N$ is the 'standard' $G_2$-invariant almost complex structure on $S^6$, then there are no nonconstant pseudo-holomorphic functions $f:U\to\mathbb{C}$ for any open subset $U\subset S^6$.