I think this is similar to Can Warding Wind block the effect of a Green Dragon's Poison Breath? but the wording of Warding Wind seems more conservative than Gust of Wind.
I just ran a combat where a druid used Gust of Wind against a green dragon. I had the dragon retaliate by flying (slowly) towards the druid and using its breath weapon. The players were delighted, immediately declaring that the monster’s attack should have been nullified because:
[Green Dragon] The dragon breathes poisonous gas in a 30-foot cone
[Gust of Wind] The gust disperses gas or vapor…
However, I ruled that the breath weapon worked as normal through the line of the wind, using similar logic to the most upvoted answer on the linked question, that "dispersing gas" is for handling cloudkill, wall of fog and other lingering effects that explicitly describe being dispersed, and that breath weapons (or other instantaneous area effects) do not get cancelled through dispersal. Instead, I would argue, breath weapons may only get blocked by spells that say it more directly e.g. Wind Wall which says that it "keeps gases at bay".
It would not have been a big deal if the players were correct in my case, as the dragon had other tactical options – moving out of the of the gust of wind to attack from the side for instance – so the ruling did not change much about the outcome (although it disappointed the players because they were convinced they had outsmarted the dragon and made it lose an attack). And in the end the battle was won by the PCs.
Should I have allowed Gust of Wind to nullify gas-based breath weapon attacks made within it, according to rules as written? What exactly does "dispersing" mean in the Gust of Wind spell description?