At best, advantage.
The poison gas doesn't require you to breath. However, as a DM, feel free to grant the Air Genasi advantage.
Note that poison gas quite often can harm people who don't breath; mustard gas, for example, causes the skin to blister (and not from acid).
Contact poisons exist. This being a dragon... we can assume that poison is poisonous in ...
No, holding your breath won't save you
The language of the ancient green dragon's Poison Breath is (MM, p. 284; emphasis mine):
The dragon exhales poisonous gas in a 90-foot cone. Each creature in that area must make a DC 22 Constitution saving throw, [...]
All it says is that there is a poisonous gas, and anyone in its cone is affected. If holding your ...