Is there anything quantum mechanical related to Fusion/Fission? (Instead of special relativity) Update : found this video of The Sun can’t work without Quantum Tunneling that expands on the accepted answer, including it for completeness.
Fusion/Fission is always accompanied by the special relativity relationship of $E=mc^2$.
Yet there are atoms that are undergoing change, and that should be in the realm of Quantum Mechanics.
Maybe in advanced quantum mechanics there are relationships that cover/relate to fusion/fission.
Special relativity (to my limited) understanding is about near light velocities, yet in a Fusion/Fission reaction nothing gets anywhere close to speed of light.
The motivation for this question came from watching a info-graphic cartoon/video on Castle Bravo and how the size of energy was miscalculated by almost 300% ,
 A: One simple example of where quantum mechanics pops up in fusion: in the sun due to the very high gravitational forces, the hydrogen nuclei are brought together. But due to the positive charges of both protons they start repelling each other. For fusion to happen the two protons have to be very close to each other so that something called the 'strong force' can bind them. But this electric repulsion does not allow these protons to be that close to each other. But, in quantum mechanics we have a phenomena called quantum tunneling, where a small particle can just 'jump' through barriers (more technically, can arrive at the other side of the barrier).
Similarly here you can think of the electric repulsion being like a barrier of energy. The protons can tunnel through them and get close enough to actually fuse. Tunneling is a very rare process, but in the sun where there exist a large amount of protons, it happens very commonly. This is how quantum mechanics plays into fusion.
A: Fusion/fission is not about atoms, but about the (atomic) nuclei. The energies involved are millions of times higher than those of the electrons bound by an atom, so resorting to the relativity becomes inescapable.
Having said that it is worth also mentioning that nuclei are essentially quantum objects, so QM is an integral part of their description alongside the relativity. The combination of quantum theory and relativity is essentially what we call quantum field theory (QFT).
