Both of these verbs are used for asking about something or looking for information. The nouns for the request / investigation are enquiry and inquiry.
Technically, enquire is used for "ask" in general ("He enquired about prices for the products"), and inquire for a more formal investigation ("The police will make an inquiry into the alleged corruption.")
However, in American English it is now very common to use inquire for everything: "He inquired about prices for the products."
When Evan asks his cellmate to look at his hands for proof that he can time travel, he then travels back to when he was a kid and stabs his palms. He now has marks on his hands as proof. But if he got those marks when he was a kid he would have had marks on his hands when he went to jail, no? So the marks being there wouldn t have proved anything to anyone but Evan himself if you think about it.