No.
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 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.
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.
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.
Yes, the familiar can use the effect of dragon's breath
The Find Familiar spell description says "A familiar can’t attack, but it can take other actions as normal". The term "Attack" means you're making an attack roll. You don't need to attack in order to use the effect of the Dragon's Breath spell (XGtE, p. 154), because there is no attack roll involved, but a saving throw instead:
You touch one willing creature and imbue it with the power to spew magical energy from its mouth [...] the creature can use an action to exhale energy of the chosen type in a 15-foot cone. Each creature in that area must make a Dexterity saving throw, [...]
Per the rules on making an attack, for something to be considered "an attack" in 5e, it must involve an attack roll or otherwise must be called "an attack" explicitly in the rules (as with grappling and shoving):
If there's ever any question whether something you're doing counts as an attack, the rule is simple: if you're making an attack roll, you're making an attack
(Player's Handbook, page 194)
you can use the Attack action to make a special melee
attack, a grapple
(Player's Handbook, page 195)
"An attack involves an attack roll or doing something that the rules call an attack, like grappling or shoving"
(J. Crawford's comment)
See also: What counts as an attack?
Since the familiar isn't making an attack, it can use the effect of the Dragon's Breath spell.
Jeremy Crawford confirms this
As @V2Blast has pointed out, an unofficial tweet from rules designer Jeremy Crawford supports this answer:
Dragon's breath is cast on a willing creature. Your familiar is a creature that can be willing. Find familiar prevents a familiar from attacking, but dragon's breath involves no attack.
Best Answer
Option 1, is the sure answer based on several considerations, especially looking at why it isn't Option 2.
It's Option 1 because:
It's not Option 2 because: