That's actually implied by 40K's old lore already, that Earth had become some kind of high tech SF utopia in line with either Star Trek or classic 60s SF, and 40K is just the result of that setting falling to decay. It was retconned all to hell eventually, and they didn't literally mean it was Star Trek, but I can see the resemblance.
There's almost nothing in the show of movies to imply that theory. Shatner really liked it though. However, Jerry Goldsmith, oddly enough, also firmly believed in it, and added a short musical cue to the first battle with the Borg that's supposed to sound like the V'Ger blaster beam from TMP. So I suppose if you want to lean on Goldsmith then the idea that V'Ger became the Borg is canon.
I personally think the theory makes zero sense. The Borg are far less technologically advanced than V'Ger is, they operate in a totally different fashion, and they're uneccesarily brutal and crude in their methods. Not really what I would expect some kind of higher being to become. Also it was implied that V'Ger had learned everything to do with reality and wanted to go beyond that somehow. It more or less left the damn universe in the movie's ending. Why would it stick around just to put implants in people? why would it even bother when it has the technology to perfectly replicate a humanoid from scratch?
I would imagine V'Ger being rather less than impressed with the Borg if it ever ran across them. You'd think a technological race descended from something that absurdly powerful would likewise be unbeatable.
I wouldn't be surprised if it was based on that hippie logic that meat is murder but its okay to eat fish because I guess our monkey brains find them less sympathetic than farm animals. Proto-veganism was a thing in the late 80s around the time the show was being filmed so it might have been a direct influence.