There are few reasons the poison damage probably does not apply to the splash damage target
Symbiotic Entity requires you to actually hit your target.
The Symbiotic Entity feature states:
Your melee weapon attacks deal an extra 1d6 poison damage to any target they hit...
The books use the term "hit" not as its standard English meaning but to describe when an attack roll exceeds or equals the AC of a target creature. The secondary effect of green-flame blade does not involve making an attack, and so you have not actually hit this second target.
- The secondary damage of green-flame blade likely is not part of the actual melee weapon attack:
The Symbiotic Entity feature states:
Your melee weapon attacks deal an extra 1d6 poison damage to any target they hit...
One way we could hope to know an answer here is by looking at the section on "Critical Hits" which states:
When you score a critical hit, you get to roll extra dice for the attack’s damage against the target. Roll all of the attack’s damage dice twice and add them together.
We know that critical hits only modify the attack's damage so if a critical hit doubled green-flame blade's secondary damage, that damage would have to be considered part of attack's damage and thus part of the melee weapon attack required by Symbiotic Entity. Unfortunately this is not very clear...
One thing we do know is that damage gated by a saving throw does not double when you score a critical hit. This is supported by this Q/A ("Is Ice Knife's explosion damage affected by rolling a critical?") and also this Q/A ("How does extra damage work for critical hits?"). This is also supported by lead game designer, Jeremy Crawford, though his idea are not official rulings:
Q. DMG Poisons: If you crit with a poison coated weapon, do you double poison dice because crit, or not because saving throw?
A. The intent is no. The saving throw, not the attack, determines whether the poison takes effect after a hit.
However, with green-flame blade there is no save that is gating the additional damage, so perhaps critical hits would double its damage like they do with sneak attack. Crawford has also had input on this specific question stating:
Q. If you critically hit with the attack part of Green Flame Blade, and are greater than level 5, do you roll extra dmg against the 2nd target?
A. The splash damage of green-flame blade isn't affected by the attack critting. Think of the attack as process X & the splash damage as Y.
Thus we at least know the intent is that green-flame blade will not double it's additional die when critting meaning it does not count as part of the attack's damage. Because of this we can conclude that it is not part of the melee weapon attack and thus cannot cause the additional 1d6 damage from Symbiotic Entity.
- Symbiotic Entity only effect a target of a melee weapon attack not all of its targets, plural.
The Symbiotic Entity feature states:
Your melee weapon attacks deal an extra 1d6 poison damage to any target they hit...
Because this feature only applies to a singular target which we hit, it cannot apply to more than one person affected by our attack. This idea also would allow us to rule that critically hitting with green-flame blade does not double the additional damage against the second target because critical hits have the same wording, using the singular form of "target".