This was an interesting question which happened last night during a game. It was a new dm so they didn't have the answer. I said it would be diluted when the gas hit the water but another player said it would go through at full power. I was just wondering if anyone knew the answer?

Per RAW, water only has an effect against fire and weapon attacks, so a non-fire breath weapon wouldn't be affected

Poison gas is definitely one a DM could rule otherwise on if they chose, but that's a slippery slope. If you decide, because of water breathing or whatever, that the gas has a reduced effect, then what about PCs that don't breathe at all like warforged or dhampir?

Interesting thoughts on this.  According to this Sage Advice from Chris Perkins, it is up to the DM's discretion.  

https://www.sageadvice.eu/how-do-dragon-breath-attacks-work-underwater/

I think there are a few questions to answer.  Is the breath weapon coming from within the water or outside of it?  Green Dragons can breath air and water, and thus if the attack started within the water I would say it would work as well as normal, perhaps even with consequences like it lingering in the medium for a round or two for reduced damage.  The poison doesn't need to be inhaled according to the attack, as it doesn't mention creatures that do not breathe, so skin contact would suffice for damage.  If the attack started from outside the water, you might rule that the range is reduced or the players in the water have something like half-cover.

For other dragons, lightning breath might have increased range, fire breath might create a fog cloud for a few rounds, etc.  Interesting things that the DM can come up with that would not be opposed to RAW.  

TLDR:  No RAW answer necessarily, up to the DM.  

DM’s call, but are some thoughts about what would happen more or less in the real world:

A) Originally green dragon breath was Chlorine gas ( now it’s just “poison gas”) - as such in water the chlorine would dissolve into the water creating HCL (or possibly Clorox bleach) in either case the damage would be converted ( at least partially) to “acid” damage rather than poison and would now linger potentially for several rounds as it diffuses away weakening slowly - possibly a 1 die/ round decrease. Ingesting any of it it would burn internally ( welcome to the burn you feel when vomiting - that HCl in your stomach) probably causing damage only healed by magic or very extended time.

B) Chlorine gas was the most common poison gas used on WWI battlefields try reading about the types of wounds it created then to see and use in describing the effects to your players. The other common gas used was mustard gas so you might want to mix the effects together.

C) realistically, because of the greater density of water it would probably shrink the range of the breath but widen its area to create something like a 30’ ball of effect that would grow as it weakens - 2x the diameter would mean 1/8 the damage. ( inverse cube  for volume changes)

D) a lot changes based on how the breath hits the water, if the breath is 90 degrees it would create a cloud above the water and a smaller area in the water with the acid/bleach effects. If it hits at an angle it’s probably going to ricochet off much like skipping a stone. If the dragon is underwater then the breath would probably really be a jet   Of acid/bleach must like a hose jet fired in a pool.

In my games, the amphibious trait and swim speed indicate that these dragons have evolved to be just as deadly in the water as on land, so the breath is unchanged. I let the players imagine it how they want unless they ask me to describe it.

Thanks  for your thoughts everyone. I must point out I'm very familiar with mustard gas as breathed in some when I was younger. The sensation of having like an invisible clamp around your lungs so you have to fight to take a breath will stay with me. Every breath gets shorter and shorter. Even to this day if I over exert myself the taste still lingers thirty years later. I only describe as so you don't have to find out what it is like.

People referencing breathing the green dragon breath like that matters. You don't need to breathe it for it to affect you. It can get you through the skin, through armour, through anything. If you're in the AoE, it affects you and nothing else matters except resistance/immunity. 

Like all dragon breaths - my points were really about the type of damage and the area of effect and how water might alter those from the standard areas and damage types. While D&D doesn’t really follow real world physics and chemistry it extract simplified for play rules that at least reference real world to some extent. That means understanding real world can help inform a DM’s decisions about how to handle situations.

D&D "poison" doesn't function like real toxins anyway.  Probably because it's not fun to bother with something that slowly incapacitates you for 12-36 hours before you die.  Or more likely you get mildly inconvenienced for the duration of the fight and then the cleric casts a 2nd level spell to cure you.

Like in any situation, if i feel when i DM that an effect could be more difficult, i may affect the roll involved, be it attack roll or saving throw. In the case of a dragon breath underwater, i could grant advantage against it for exemple.