If you are talking about labelled graphs (distinguishable vertices) then the fact that these are graphs is irrelevant. So you are really asking if there is a Hamilton path through all the subsets of an arbitrary $\binom{N}{2}$ set where each move adds or removes an element.

This is just a Hamilton path in the $\binom{N}{2}$-cube which certainly exists and is well known.

Look up “Gray code” for details.