If $G:E\rightarrow F$ is a holomorphic bundle map over a smooth map $f:M\rightarrow N$ where $M$ and $N$ are two complex manifolds, there is a result that say that every local smooth section can be write as a combination of holomorphic sections. So, if $\{e_i\}$ and $\{f_i\}$ are basis of holomorphic sections of $E$ and $F$ respectively we have, for all$x\in M$, $G(e_j(x))=\alpha^i_j(f_i(f(x)))$ for same functions $\alpha^i_j:M\rightarrow C$.
How can I show that this $\alpha^i_j$ are holomorphic functions?
Thanks for all help me.
Let me rewrite your actual question so it makes sense:
Suppose that $\xi: F\to N$ is a holomorphic vector bundle of finite rank $n$. Let $U\subset N$ be an open subset over which the bundle is holomorphically trivial, i.e. the restriction of the sheaf of holomorphic sections of $\xi$ to $U$ is free over the ring of holomorphic functions ${\mathcal O}_U$ with the basis $f_1,...,f_n$. In other words, the restriction $\xi_U$ of $\xi$ to $U$ is (holomorphically) isomorphic to the trivial bundle $U\times {\mathbb C}^n$ where the sections $f_1,...,f_n$ become the maps $$ w\mapsto (w, \pi_i), i=1,...,n, $$ where $\pi_i$ is the constant map (identically equal to $1$) to the $i$-factor of ${\mathbb C}^n$.
Suppose that $h$ is a smooth section of $\xi|_U$, i.e. a smooth map $$ w\mapsto (w, \phi(w))\in U\times {\mathbb C}^n, $$ i.e. $\phi=(\phi_1,...,\phi_n)$, where $\phi_i$'s are smooth complex-valued functions on $U$. (You are also assuming that $h$ comes from some smooth morphism of holomorphic vector bundles which is an irrelevant piece of information.)
You are then asking if there are smooth complex-valued functions $\alpha_1,...\alpha_n$ on $U$ such that $$ h(w) =(w, \sum_{i=1}^n \alpha_i(w) \pi_i(w)). $$ The answer is trivially positive: Take $\alpha_i=\phi_i$, $i=1,...,n$.