I'm willing to bet that in this question, $G$ is a cyclic group of order $N$. To that end, I'm answering the question I think you meant to ask. If I'm wrong, please let me know. In the case where we have a cyclic group of order $N$, there are $\varphi(N)$ different generators, each of the form of the question.
To answer your question: No. When we say an element (say $g$) generates a group, it means that every element of the group is a power of that element (of the form $g^n$ for some $n$). If we were to say that the two elements $g,h$ generate $G$, often written $G = \langle g,h \rangle$, we mean that any element of $G$ can be written as $g^n h^m$ for some $m,n$.
So any two generators generate the same set of elements. By distinct generators, we mean that the two generators are different from eachother. For example, if we are working with the cyclic group of order $2$, there are two elements. We might call them $e$ (the identity) and $g$, so that $g^2 = e$. But $g^3$ also generates this group. But $g^3$ is the same as $g$, so they are not distinct even though the 'words' describing them are not the same.