The gust causes the gas to vanish in the 10-foot-wide column.
The gust of wind spell lists the following effects, among others:
A line of strong wind 60 feet long and 10 feet wide blasts from you in a direction you choose for the spell’s duration. Each creature that starts its turn in the line must succeed on a Strength saving throw or be pushed 15 feet away from you in a direction following the line.
The gust disperses gas or vapor, and it extinguishes candles, torches, and similar unprotected flames in the area. (…)
A strong wind is defined in the Dungeon Master’s Guide as follows (p. 110; emphasis mine):
A strong wind imposes disadvantage on ranged weapon attack rolls and Wisdom (Perception) checks that rely on hearing. A strong wind also extinguishes open flames, disperses fog, and makes flying by nonmagical means nearly impossible. (…)
As “disperse” isn’t a defined game term, and D&D 5e is written using natural language, we need to refer to the normal meaning of the word. Merriam-Webster includes the following definition:
- a: to cause to break up (see break up sense 1a)
- police dispersed the crowd
- b: to cause to become spread widely
- c: to cause to evaporate or vanish
- sunlight dispersing the mist
In particular, 1c is the most relevant definition, dealing with “disperse” as it relates to an environmental effect (mist). So we have to conclude that the meaning of “disperse” in the spell description, as it applies to a gas in this context, is to cause the gas to “vanish or evaporate”.
As a result, the gust of wind spell causes the gas to vanish in the 10-foot-wide column, and thus has no effect on the creatures behind the line of wind. This does not mean that the breath weapon has no effect on creatures on either side of the column (if its cone is appropriately wide).
This DM would take account of the dragon’s positioning to determine just how much of the breath weapon is dispersed. For example, if the dragon releases its breath weapon while its mouth is directly in the gust, then this DM would rule that the gust nullifies the whole breath weapon (as it vanishes before it has a change to “fill” the cone). Alternatively, were the dragon to release its breath weapon from above the gust, angled downwards, then only the central line of the cone would be dispersed (protecting creatures behind the gust).
As an aside, if the dragon is flying, within the column of the strong wind (as you indicate), then it must either land at the end of its turn or fall (and take falling damage) due to the Strong Wind rules. It also must spend 2 feet of movement for every 1 foot it moves toward the caster (in effect, difficult terrain) due to the gust of wind spell (on top of the Strong Wind effects).