If you enquire about someone or something you ask about them.
She enquired about his health. He was touched by her enquiry..
In British English we use inquiry when we’re talking about more official investigations.
She thought the committee had no right to inquire into her politics.
There was an inquiry into the death of Ian Tomlinson.
!Note – If you can’t decide, stick to inquire / inquiry. You’ll sound like an American, but that can’t be helped.