Ok, not really a "faction"; more of a proof of concept type thing. But anyway, I was playing around in the costume editor the other day and realized you can make a pretty decent looking Borg with with basic parts:https://imgur.com/a/DKwBS
So I thought, Ok; I can make my character and crew look like Borg. I can make my ship look like it's been assimilated (using Borg rep gear), so if there were just Borg themed missions (with you playing that role) you would basically have a Borg faction.
With those thoughts in mind I created the foundry mission called "Borg Faction Test", which you can currently find in the "review" section for both Fed and KDF characters.
If you decide to play, I highly recommend making your character over like the example I posted above and using the Borg parts on your ship, to get the right feel and atmosphere.
I'm not saying this mission makes the case that there "should" be a Borg faction; but maybe it at least gives an idea of what one might feel like.
Addendum: here is something I could realistically see happening one day. A Borg "faction" where your tutorial is you as a Fed/KDF responding to a distress call. A base, or something, is under attack by the Borg. After a valiant effort, the battle is won...by the Borg. You are assimilated. Then you play through half a dozen episodes wreaking havoc as a Borg, after which you are eventually rescued and continue to play afterwards as a Liberated Borg.