Just a strange off-the-wall thought I had.
If I'm the leader of a powerful alien race who needs earth for some sort of intergalactic resource base, I'm going to teleport one of my guys into the middle of the GB's Wednesday morning meeting and tell them he's a messenger of Jehovah and that destruction of "Satan's world" is imminent. The Borg would have around 6.5 million ready slaves to do the grunt work and would all shout with glee at the genocide of the nearly 7 billion other people on the planet and the prospect for slaving for this alien race for 1,000 years.
They're ripe for the picking. Just teleporting into the locked room should be enough to convince them but you could blind them with science (fake miracles) if any are doubting Thomases.
Now that my mind's running with this idea, how do we know the GB isn't already infected by alien control? Perhaps in some other universe it already is...