Is there anything quantum mechanical related to Fusion/Fission? ( Instead of special relativity)

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.