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 $U\subset S^6$.

**Added in response to the OP's comment:**

Here is how you can see how the Nijnhuis tensor induces restrictions on the possible first derivatives of a pseudoholomorphic map: Suppose that $(N,J_N)$ and $(M,J_M)$ are almost-complex manifolds and one wants to study the conditions on a pseudoholomorphic mapping $f:N\to M$ that satisfies $f(y) = x$ for some $y\in N$ and $x\in M$. Choose a basis $\alpha^i\ (1\le i\le m)$ for the $(1,0)$-forms on an $x$-neighborhood $U\subset M$ and a basis $\beta^p\ (1\le p\le n)$ for the $(1,0)$-forms on a $y$-neighborhood $V\subset N$. (Assume that $U$ and $V$ have been chosen so that $V$ is in the domain of $f$ and $f(V)\subset U$.) Then the desired mapping $f$ will satisfy $f^*\alpha^i = F^i_p\ \beta^p$ for some functions $F^i_p$ that are essentially the components of the Jacobian of $f$ relative to the bases $\alpha$ and $\beta$. Now, there will be unique functions $A^i_{\bar k\bar l} = -A^i_{\bar l\bar k}$ on $U$ and $B^p_{\bar q\bar r} = - B^p_{\bar r\bar q}$ on $V$ such that
$$
\begin{aligned}
\mathrm{d} \alpha^i
\ &\equiv \tfrac12A^i_{\bar k\bar l}\ \overline{\alpha^k}\wedge\overline{\alpha^l}
\ \mod \alpha^1,\ldots,\alpha^m \\
\mathrm{d} \beta^p
\ &\equiv \tfrac12B^p_{\bar q\bar r}\ \overline{\beta^q}\wedge\overline{\beta^r}
\ \mod \beta^1,\ldots,\beta^n
\end{aligned}
$$
(These functions are the components of the Nijnhuis tensors of the two almost-complex structures relative to the chosen bases of $(1,0)$-forms; they vanish identically if and only if $J_N$ and $J_M$ are integrable on $V$ and $U$.) Now, since $f^*$ preserves $(p,q)$-type and commute with the exterior derivative, it follows that, upon taking the exterior derivative of the relation $f^*\alpha^i = F^i_p\ \beta^p$ and comparing the $(0,2)$-components, one has
$$
f^*\left(\tfrac12A^i_{\bar k\bar l}\ \overline{\alpha^k}\wedge\overline{\alpha^l}\right)
= F^i_p\ \tfrac12B^p_{\bar q\bar r}\ \overline{\beta^q}\wedge\overline{\beta^r},
$$
which implies
$$
A^i_{\bar k\bar l}{\circ}f\ \overline{\left(F^k_q\ \beta^q\right)}\wedge
\overline{\left(F^l_r\ \beta^r\right)}
= F^i_p\ B^p_{\bar q\bar r}\ \overline{\beta^q}\wedge\overline{\beta^r}
$$
i.e., one has the algebraic equations
$$
\tfrac12\ A^i_{\bar k\bar l}{\circ}f\ \overline{(F^k_qF^l_r-F^k_rF^l_q)}
= B^p_{\bar q\bar r}\ F^i_p\ .
$$
In particular, since $f(y) = x$, one has
$$
\tfrac12\ A^i_{\bar k\bar l}(x)\ \overline{(F^k_q(y)F^l_r(y)-F^k_r(y)F^l_q(y))}
= B^p_{\bar q\bar r}(y)\ F^i_p(y)\ .
$$
Of course, these relations are trivial if the respective Nijnhuis tensors vanish at $y$ and $x$, but they can easily be quite restrictive. For example, if $J_M$ is integrable, so that the $A$s vanish, then this is a set of linear equations for $F(y)$ that, depending on $B^p_{\bar q\bar r}(y)$, could easily have only $F(y)=0$ as solutions.

In fact, as soon as the dimension of $N$ is at least $3$, the 'generic' Nijnhuis tensor $B$ will have this property, and a conclusion will be, for example, that, for the 'generic' almost-complex manifold $(N,J_N)$ of dimension $3$ or more, there are no nonconstant pseudoholomorphic functions $f:V\to\mathbb{C}$ for any open set $V\subset N$. (The conclusion holds in the case $\dim N = 2$ as well, but you have to differentiate one more time to get this in the $2$-dimensional case.)