V'ger was actually linked to the Borg by the Great Bird of the Galaxy himself. Roddenberry implied a connection in an interview IIRC regarding TNG and the episode 'QWho'. It's obvious STO has paid homage to this with the sector block Red Alert events. EDIT- Although it is possible Roddenberry was joking in that interview.
Remember TMP was pretty vague about it. Spock had a brief mind-meld, it's entirely possible he only got a fraction of V'ger's history or only parts of it could be interpreted. A planet full of 'living machines' could easily be interpreted as a planet full of Borg. We don't know, because we never see it for ourselves.
To address the OP, I don't think Q or the Continuum had anything to do with the formation of the Borg. But it is one of Star Trek's biggest mysteries. Among many, really.
...Spock had a brief mind-meld, it's entirely possible he only got a fraction of V'ger's history or only parts of it could be interpreted. A planet full of 'living machines' could easily be interpreted as a planet full of Borg...
Assuming the Borg did create V'Ger, I think this is reasonable considering Spock pretty much went into shock due to information overload.
Or maybe the Pakleds were messing around with a toaster one time and things got out of hand...
I doubt any connection between V'ger and the Borg. For one thing, V'ger considered all organic life as "infestations" and machines as the only true lifeforms, whereas the Borg deliberately seek out life to assimilate, then there is the incongruity of the detailed elegance of the Ilia probe compared to the haphazard and clunky functionality of Borg drones, if the Borg came from V'ger then something went horribly wrong between the time of their creation and the first time the Federation encounters them.
I've always envisaged the Borg as the result of an experiment in nanotechnology that went horribly wrong, or the creation of some mad scientist type that let loose her creation on an unsuspecting populace, becoming the first queen and template for those who followed.