For most gasses no, but it ultimately depends on the poison
In standard English, one might expect 'fresh air' to mean free of poison.
However, the Spelljammer rules (from which the Air Bubble spell comes) define 'fresh air', meaning that in this case it is a game term, and the standard English definition does not apply.
In Chapter 2, "Air Quality", we are told that fresh air is defined as "completely breathable". This contrasts with deadly air, which is unbreathable and causes suffocation. In between is foul air, which instantly imparts the poisoned condition without a save.
Thus, the Spelljammer rules are telling us that "fresh air" contains whatever unspecified substance1 creatures need to breathe (and fire to ignite), while whatever alternate substance is in foul and deadly airs is a poison. Within this context, the Air Bubble spell, by specifically creating and maintaining 'fresh air', works against the specific 'poison' found in foul and deadly airs.
Now consider a Spelljamming vessel, with its supply of fresh air. Were someone to release a poison into the air, such as a cloudkill spell, would it no longer be fresh air? Would it foul the air of the ship? No, because even though the air would be poisonous, it would be just as breathable, in the sense that creatures breathing it would take damage, but not suffocate. Because the cloudkill effect does not cause suffocation, the air of the ship would still be considered fresh, within the Spelljammer definition of fresh air.
As an analogy, consider 'fresh water'. I could add poison, or harmless dye, and it would still be fresh water. It would only become 'salt water' if I added salt. Similarly, fresh air (within the context of Spelljammer) is not fouled by just any poison, but only by the specific poison that causes deadly air to be suffocating.
Now that we understand the nature of fresh air, will the Air Bubble spell keep out poisons? No, because poisonous gasses do not cause suffocation - rather they cause poison damage or the poisoned condition. These poisonous gasses would diffuse into the space of the Air Bubble and do their thing without reducing the breathability of the air, which is the only thing that the spell is maintaining, much like the mercaptan from a leaky gas line could diffuse from the kitchen into our bedroom - we would smell it, even though we were still in fresh air and able to breathe.
But wouldn't the spell's globe keep the poison gas out by acting as a physical barrier? Perhaps if it were a solid helmet, but it is specifically called "spectral". There is no 5e definition of this word, but it is typically used for 'quasi-real' spell effects that can exert forces but which themselves can't resist things moving through them (with the exception of guardian of faith). Thus, absent explicit direction that the globe blocks the movement of gasses, it appears to be merely cosmetic, serving to delineate the area of effect of the fresh air.
Some monsters, such as the water elemental, have attacks that suffocate or prevent characters from breathing - the Air Bubble would guard against that, but that is not the poison gas the OP asks about. Other monsters, such as a green dragon2, produce poison that damages the creatures which breathe it, but does not suffocate them. Again, the Air Bubble will be ineffective against poisons that don't work by suffocation but merely by poisoning other parts of one's body.
However, the Air Bubble would protect against poisons which function by specifically keeping the target from being able to breathe (in real world terms, obviously carbon dioxide, but also things that interfere with the uptake of oxygen like carbon monoxide and hydrogen cyanide). That is, if there was an in-game gaseous substance that either gave the poisoned condition or did poison damage, and also suffocated a creature or removed its ability to breathe (as the foul and deadly airs explicitly do), then Air Bubble would protect against that thing.
A quick look through the inhaled poisons in the DMG and some obvious spells doesn't present any outstanding candidates for poison damage by suffocation. However, such an effect could be out there or added later.
1Typically 5e rules gloss over most aspects of real-world biochemistry. The 2e Spelljamming rules, in contrast, actually used the terms "oxygen" and "carbon dioxide" and treated carbon dioxide as a poison.
2Prior editions specified that green dragons breathe chlorine gas. As above, 5e has set aside the specifics of the gas and just calls it "poison".