Torsion-free quotient of a globally generated vector bundle

Question

Let $E$ be a vector bundle over a smooth projective surface over the field $\mathbb{C}$. Suppose $E$ is globally generated and $N$ is a torsion-free quotient of $E$. Is $N$ also globally generated? I can see that there is a composed surjection $$H^0(E)\otimes O_X\rightarrow N\,.$$ But how do we show that there is a surjection $$H^0(N)\otimes O_X\rightarrow N$$?