'Pick a fight' is the correct term. 'Pick up a fight' makes no sense.

You can "pick up where you left off" So if you want to say this: "Pick up the fight where you guys left off!" You can. But simply saying "pick up a fight" makes no sense.

You've confused the two definitions of pick :) Pick can mean to poke/scratch at something. It could also mean to lift/carry something up. To pick a fight is figurative language but the first definition is still correct.

"Pick up a fight" doesn't exist as a fixed expression. "Pick a fight" is correct