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.

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 $k$-subsets of an arbitrary $\binom{N}{2}$ set where two sets are adjacent if their symmetric difference has size two.

These are many such Hamilton paths, even Hamilton cycles, known under the general term "combinatorial Gray code".

Look up Carla Savage's survey of combinatorial Gray codes.