Who would win?

I don't know that Zorg. I only know this Zorg.
- No matter how many times I see that movie, I'm always surprised like a toddler playing peek-a-boo when I realize (yet again) that is Gary Oldman. He is as unrecognizable in that role as he is fantastic.
  
  Truly a master chameleon.
- Jean-Baptiste Emanuel!
- So sad how dogshit imgur has become.

Borg assimilate Zerg.
BORG RUSH
Game Over
- The zerg would be difficult to assimilate though, they have such a short incubation period, and we know the Borg implants are added after birth. logistically it's complicated to manage.
  I don't think they'd be worthy for assimilation as a whole either, as the underlings couldn't't meaningfully contribute to the Borg hive mind.
  
  If they assimilated the hive mind then the swarm would be under their control
- They would also assimilate the Protoss as well... (and humans, not that it matters as Borg are already far beyond nukes)
  We will add your biological and technological distinctiveness to our own... resistance is futile...
- The Borg couldn't even assimilate 8472 aka the Undine. The Zerg make those bastards look like baby's first eugenics war. And thats not even factoring in Kerrigan's BS writing.

Kerringan has this ability where the entire plot bends ass-backwards to make her come out on top. Can't beat that.

On the ground, Zerg, they move too fast. In space, the Borg. A cube probably shrugs off whatever organic projectiles the Zerg use.
- The Borg's real danger is their nanoprobes, not the drones. The Borg have shown they are not afraid to experiment with alternative delivery methods up to and including planetary dispersion bombs lol
- Scourges would disagree. Also Leviathens are bigger than Cubes.

What's the Vespene Gas situation?
- You require more.
  
  Damn. When will the amount of vespine gas I have be enough?

Borg all the way, all the time. The Zerg would never rate a two part season ending cliffhanger.
- What about two entire expansion packs? Borg never got that!

I get the feeling that the Borg would think they had won, but it turns out the Zerg Hive Mind infested the Borg queen, so now the Borg are Zerg...

- Borg Queen, Borg Queen, Borg Queen, Borg Quee-een
  I'm beggin' of you, please don't take my Cap't
  
  standing there with just a head
  \ Hearing every word that's said
  \ Nothing here to stop her Galactic plan
  \ Then a starfleet captain came
  \ And the delta quadrant was not the same
  \ Because they feared the wrath of Katherine Janeway!

Assimilating a single Zerg doesn't change the fact there is 10,000,000,000 more of them rushing in.
- Okay, but The Borg have comparable numbers. Humans are Species 5618 when the Borg make first contact. That's 5617 species they have assimilated (or, at least, tried to assimilate) beforehand.
  In Voyager, the whole Delta Quadrant is teaming with Borg. Functionally on par with the Zerg.
  
  Identified for sure, not necessarily assimilated. Can't remember who the moment but there are people who the borg ignored because they were uninteresting. But there are species the Borg have identified and not assimilated as well
  
  Borg can’t match the lifecycle speed of Zerg though. Yes, they can create new infants, implant them, and accelerate their growth in maturation chambers to produce new adult humanoid drones, but the process took 17 cycles (days). Zerg on the other hand produce new drones in a matter of seconds and new hatcheries in a minute or two. Within a single day they could multiply across an entire planet or overwhelm a cube.

I would win
- Confirmed!!!

I don't think either would win, the Borg would find them not worthy of assimilation and retreat or destroy whatever planets became infested with zerg.
You can't add implants to every new hatched zergling, and they offer no technological distinction.
Overlords maybe can be assimilated, but it's not clear if that assimilates their eggs.
- We will add your biological and technological distinctiveness to our own. The adaptability alone would probably be very handy for them, instead of having to deal with whatever random species' drones they have laying around, they could specialize much more efficiently. They were interested in 8472 for exactly that reason IIRC. And it's not like it is a hassle to add the implants, they have all kinds of nanite delivery systems. If the eggs were produced from an assimilated source then they would presumably already be implanted before being laid. But it wouldn't be surprising to see them produce some kind of industrial hatchery that automatically implanted them if they had to come from unassimilated hosts for whatever reason.
  
  I thought they wanted 8472 because they were immune to most weapons, had superior weapons to the borg, and had an extremely advanced immune system, to the point it broke down their nanites.
  
  Zerg are pretty much useless to the Borg though because their adaption is entirely reliant on the knowledge and memories of the assimilated. They took over entire worlds in pursuit of knowledge to subdue 8472. And failed at every attempt in alpha canon.
  The Zerg use a psionic hivemind, the vast majority of Zerg strains are not sentient, or effectively puppets for the sapient commanders of the swarmz usually cerebrates and later broodqueens, even things live Overlords are just relays.
  The Borg have never been shown capable or willing to apply genetic modifications to drones, such as enhancing human drones with the physical abilities of vulcans or klingons that the nanites and implants don't already imbue. Since Zerg evolution and mutations aren't tech based, the borg can't replicate it, which was their goal for 8472 in being able to use that tech for their own ends.
  Borg ships also don't have shields, so it's just going to be drop pods infesting cubes and eating them inside out for biomass and infesting the rest of the structure for use as mobile hive clusters. It the 8472 were able to overcome nanites with their own immune systems, the Zerg would equally be able to become immune to such attacks particularly if Abjuthar had any part in it.