No, it is not possible unless the $f_i$ are all constant.

Take a polynomial $p$ of $m$ variables. $g(z) = p(f_1(z), \ldots, f_m(z))$ is a holomorphic function on $\Omega$, and generically non-constant if $f_1, \ldots, f_m$ are not all constant (if we happen to pick a $p$ for which it is constant, and $f_i$ is non-constant, then replace $g(z)$ by $g(z) + t f_i(z)$ for any $t \ne 0$). By adjusting the constant term of $p$, we can ensure that $g(z_0) = 0$ where $z_0$ is some given point of $\Omega$.

EDITED:

No, it is not possible if $f_1, \ldots, f_m$ are not all constant.

Suppose wlog $f_1$ is not constant. I'll ignore $f_2, \ldots, f_m$ and consider polynomials $p(f_1(z))$. For convenience I'll omit the subscript and call this $p(f(z))$. $f(\Omega)$ is an open subset of $\mathbb C$, so it contains $\alpha + \beta i$ for some $\alpha, \beta \in \mathbb Q$, i.e. there is some $z_0$ such that $f(z_0) = \alpha + \beta i$, and so $p(f(z_0)) = 0$ where $p(w) = (w-\alpha)^2 + \beta^2$.