As an edge case, the RAW technically allows this

You can use the poison in this vial to coat one slashing or piercing weapon or up to three pieces of Ammunition. [...] A creature hit by the Poisoned weapon or Ammunition must make a DC 10 Constitution saving throw or take 1d4 poison damage.

The problem here is that this is an edge case. Normally, a slashing or piercing weapon is a weapon that does only slashing or piercing damage. Nowhere in the rules does it explicitly say that the poison must be delivered by those damage types. But since the rule only mentions the weapon and a halberd is a weapon that does slashing damage, technically the poison could be applied here even though the Polearm Master attack itself only does bludgeoning damage.

Logically, this shouldn't work

However, if we look at the rules from the perspective of what they are trying to say instead of the strict letter of the rules, I think we come to a much more reasonable ruling.

The rules state that a slashing or piercing weapon is what needed to deliver the poison. In the rules, damage types are almost always attached to weapon damage and not weapon identity. So, this wording is very unusual. Even so, the damage type is clearly a reference to the expected damage type of the weapon damage. Thus, I think it is highly reasonable to read the poison rules as saying "slashing or piercing damage from a weapon" because of this.

In fact, one of the sample poisons from the DMG (published after the PHB where the Basic Poison is written) is already written this way¹:

A creature that takes piercing or slashing damage from an object coated with the poison is exposed to its effects.

Narratively this all makes sense as logical inference from the rules seems to be that the target must be cut or pierced in order for the poison to enter the body and take effect.

^{1 - Thanks @NautArch}

There are few reasons the poison damage probably does not apply to the splash damage target

1. 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.

2. 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.

3. 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".

A note: This does of course require that we assume the phrase "any target" is, in fact, singular. As user @thedarkwanderer has pointed out this phrase could be read as already being plural in which case my argument for point 3 would not apply. There's ambiguity in the meaning as the word "any" in English - It can be used as the plural over the alternative options of "each/every".

A small note on the exceptional state of green-flame blade:
This spell is rather exceptional; you make only one attack but end up affecting multiple creatures. The Sage Advice Compendium has said the following regarding the Ranger's Whirlwind Attack and Volley features:

Whirlwind Attack is unusual, in that it’s a single attack with multiple attack rolls...

These two features and green-flame blade are the only cases I know of where a single attack affects multiple creatures and so they probably have some odd interactions such as the one asked about in this Q/A ("How does the Sharpshooter feat interact with the Hunter ranger's Volley feature?")