wife问：你为什么不关心儿子的学习？ 丈夫答：我没有关心儿子的学习是因为我工作太忙了。

I agree with Lee mostly. "不" carries a few subjective feeling like "我今天不(dont want to)做饭" ."没有" is stating a fact like "I didnt cook today. following is 3 examples for a better understanding 1) 他父母不关心他= his parents dont care about him 2)他父母没有关心他=It's stating a fact that his parents dont care about him 3)他父母没有关心过他= his parents have never care about him

The way it appears to me is that 关心 is a verb meaning “to be concerned with" or "to care about." So, to say "I am not concerned with X," you would use 不关心. Since 不 negates verbs, this is appropriate. However 没有 literally means "not have" and is used to negate nouns, so 没有 would not make sense when paired with 关心.

你可以说 ：我没有人关心。 you cannot miss the 人. that means: No one cares about me. It the only case I can retell about the 没有关心。 but you can use 没有关注 e.g. 我对这件事没有关注。 that means I am not notice about this things.