No.
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The key factor is:

>  It hedges out vapor, **gas**, and fog **that can be dispersed** by strong wind. 

So if the effect description of said gas includes a note that it can be dispersed by a strong wind, it will be.

The green dragon breath has no such statement, therefore it is not warded by Warding wind.

OP asked in comments if a stronger wind would affect the Green Dragon's Breath:

It doesn't matter the wind. The breath of the green dragon would spread out to its cone-shaped area of effect even inside a hurricane. Unless an effect's description states that no gas can get through or something amounting to that, it doesn't matter. That spell cannot stop the breath because the breath cannot be dispersed by wind (and the spell states that it only affects what "can be dispersed"). The dragon entry would have "can be dispersed" if it would.

Regarding wind as weather in the DMG, a strong wind would not affect the Dragon's breath, because it is **gas**, not **fog**:

> STRONG WIND
>
> 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. A flying creature in a strong wind must land at the end of its turn or fall.

User [Doval in the comments](https://rpg.stackexchange.com/questions/104768/can-warding-wind-block-the-effect-of-a-green-dragons-poison-breath/104771?noredirect=1#comment251046_104771) below brought an example of a higher (3rd) level spell without the **that can be dispersed** limiter:

> (...) Wind Wall also creates a strong wind and says: "The strong wind keeps fog, smoke, and other gases at bay" which RAW would keep out a green dragon's breath. (...)

An effect that stated "blocks gases" without the "*that can be dispersed*" remark of Warding Wind would be effective to block the Green Dragon's Breath.

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Example of an effect that blocks gases without the "*that can be dipersed*" clause:

> **Cube of Force**
>
> Face 1, 1 charge: Gases, wind, and fog can’t pass through the barrier.

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Example of an effect that is hedged out by warding wind:

**Fog Cloud**

> You create a 20-foot-radius sphere of fog centered on a point within range. The sphere spreads around corners, and its area is heavily obscured. It lasts for the duration or **until a wind of moderate or greater speed (at least 10 miles per hour) DISPERSES it**.