How powerful do you think the borg would get if they got thire hands on a Q?
The Voyager series introduces us to a multiverse. Every multiverse has a 'Frequency' in some kind of subspace spectrum. If you can get yourself to change to that 'Frequency' you switch universes. This is the explanation given for how occasionally a transporter accident sends people to alternate universes.
In at least one of these universes the Borg defeats the Federation during its formative period, and the Commander Riker from that universe is both a coward and a traitor. The character of Starfleet is its bond which is what results in its strength. That is a constant in all Star Trek up to and including Deep Space 9. They will choose destruction over corruption of Starfleet's character. The exception is this other universe in which Riker is a coward and the Borg have overrun Federation space.
Now let us suppose that the Q have a similar weak Q, like Riker, in at least one of these other universes. Maybe this Q goes crazy or becomes a traitor or has a love for the Borg or becomes so bored that they decide to try giving the Borg power. This would not be enough to change things for the reason that there is more than one almighty Q.
Each Q is almighty. Therefore any one Q is able to match and oppose and counterbalance all of the others. The only way that the Borg could benefit from capturing one of them would be to capture all of them. If somehow that happened it would mean that all of the Q's had decided to join the Borg. In that case the future would be Borg, however I cannot think of a reason for all of the Q to do that. Any one of them is many times more powerful than the Borg; and joining the Borg would also not relieve their boredom. More than likely it would make their boredom worse. The Q have commented before that the human race is interesting. This precludes the possibility of them choosing the Borg.