You seem to be asking two things, could a black hole's magnetic fields cool nearby matter, and could this cooling produce cold fusion. But maybe we should first ask whether "cold fusion" is a real thing.
Nuclei contain protons and neutrons held together by pions. Fusion is when two nuclei become one. The barrier to this happening, is the positive electric charge of the protons. The pion force is very short range, and the electric repulsion between protons makes it difficult for protons to get close enough for the pion force to dominate.
Hot fusion overcomes the electric repulsion because the nuclei are colliding very hard in the hot plasma. If they come right at each other with enough kinetic energy, they can overcome the repulsive force and get close enough for the strong attractive pion force to trap them together.
Cold fusion is the idea that nuclear fusion can occur without high temperatures and high-energy collisions everywhere. But then the problem is, how do the nuclei get the energy to overcome the repulsion of their electric charge, and get close enough to fuse? The energies in chemical reactions are nowhere near enough.
This difference in energy between chemical and nuclear reactions is why most physicists don't believe in cold fusion, and it's why "cold fusion theory" is so full of people who want to overturn a lot of basic modern physics. From a conventional physics perspective, the only reasonable theory of cold fusion that I ever saw was posted by Ron Maimon on this site a few years ago. It was intricate and ingenious and involved highly specific properties of palladium and deuterium, so even if true, it wouldn't be a process that could generalize easily to other atoms, compared to the hot fusion method of "have them collide really hard".
I can think of one other type of ultra-cold fusion that might work, and that would be through Bose condensation. This is a quantum effect in which particles get into the same state. It only works for some types of atom and requires extremely low temperatures, and is very unstable. It might be effectively impossible to make an atomic Bose condensate dense enough for fusion to be likely, it might inevitably destabilize long before that.
As for whether a black hole's magnetic fields could cool its environment, it seems a somewhat arbitrary idea. The magnetocaloric effect requires a magnetizable substance that will then cool something else, and it also requires the magnetic field acting on the substance to vary over time in the right way.
It's a bizarre vision to imagine e.g. a black hole surrounded by a disk of deuterium and metallic dust, the dust driven through a cycle of magnetization and de-magnetization by fluctuations in the black hole's magnetic field, and thereby cooling or supercooling the deuterium plasma, with some type of "cold fusion" occurring in this plasma...
Of course, modern astrophysics says the universe contains many phenomena which are just as amazing as that would be. For me, maybe the bottom line is that, even if some version of your scenario is physically possible, I see no reason why it would be normal or common or even possible without very special circumstances, e.g. it might be artificially created by an alien race.