I think $G$ is supposed to be bounded and hence $\overline{G}$ is compact. By continuity of $|f|+|g|$ and compactness of $\overline{G}$, there exists $z_0\in\overline{G}$ and $M\ge 0$, such that $|f|+|g|$ attains its maximum $M$ at $z_0$. Let $a,b\in\mathbb{C}$ with $|a|=|b|=1$, such that $|f(z_0)|=af(z_0)$ and $|g(z_0)|=bg(z_0)$. Define $h=af+bg$. Then $h$ is holomorhic on $G$ and continous on $\overline{G}$. Moreover, $|h(z)|\le M$ on $\overline{G}$ and $h(z_0)=M$. Then the conclusion follows from maximum modulus principle.