I just watched Star Trek: First Contact again, and then I reflected back on Voyager. What's going on there? Are the Borg gone? Is the Queen finally dead, or is she forever lurking?
There are only a handful of major Star Trek stories that have taken place after the the last episode of Voyager (Endgame):
- Star Trek: Nemesis - No Borg
- Star Trek: Countdown - This was a canon comic book mini-series published as a lead-in to the 2009 Star Trek reboot. It takes place in 2387, 9 years after Endgame. In it, you see the supernovas endangering Romulan Empire and the efforts of Spock to use the red matter to help solve the problem. In particular, the Romulans enhance the Nerada (Nero's ship) with Borg technology, implying that, at the least, some Borg technology remained functional even after the deployment of the neurolytic pathogen.
- Star Trek Online - This MMORPG takes place in the prime timeline in 2409, 22 years after the destruction of Romulus. One of several storylines is the Borg reappearing after more than 30 years of quiet post-Endgame.
So how you answer the question depends on how you interpret the canonicity of Countdown and Online.
Since we now have another Star Trek series, I'll go ahead and add this answer:
The Romulans state that, to the collective, the Artifact (damaged Borg Cube) is a graveyard, implying the collective's continued existence. At time of this writing, I've seen up to S1E5, so there maybe more on this subject later.
Picard continues to speak of the Borg in the present tense, and we see new tech that the Borg assimilated, which featured in a Voyager episode. I read one article (Screen Rant?) that argued that the Queen used it to survive the events of Endgame (not the Avengers version!), although how she managed it lying in pieces on the floor is something the article neglects to mention.
Also, the Memory Alpha page for a spoiler ship states that
It was assimilated sometime in the 2380s, which is at least two years after Voyager's return. The Borg must have lived.