They’re different things. The OP means electromagnetism, Coulomb’s law has nothing to do with quantum mechanics, it’s classical physics.
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The relation between them is that they’re both forces that scale with the inverse square of the distance between the objects. Any force that scales with the inverse square of distance has pretty much the same general form.
Another similarity is that both are incomplete, first approximations that describe their respective forces. The more complete versions are Maxwell’s laws for electromagnetism and General Relativity for gravity.
It’s electromagnetism you mean, not quantum mechanics.
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