Dragons do need to breathe (note that black dragons have the ability water breathing, indicating that a dragon without it can't breathe under water and is subject to drowning rules), so anything that asphyxiates them will kill them, regardless of the element. A dragon's resistance to damage of certain elements doesn't save them from needing to breathe, just from being directly injured by that element.
However, there is no information on how long a dragon can hold its breath, so how long it takes to drown a dragon is up to the DM's discretion. An easy answer is that you would just use the normal drowning rules, so how long it takes depends on a dragon's Constitution score, but there's nothing to stop a DM from house-ruling that dragon physiology works differently in their game's setting.