Besides the great answers that have been given here, I'd add that question can be both a noun and a verb while we'd use enquiry for the noun and enquire for the verb. Hope this helps! :)

An enquiry is a polite question. "I have an enquiry about your prices." An inquiry is a legal investigation, such as "The Grenfell Tower Inquiry".

(I'll use the U.S. spelling: "inquiry") The difference is that an "inquiry" is intrusive, whereas "question" is not. Here's an example. Suppose you are walking past a clothing store and you wonder: "Does this store sell mens' hats?" That is a question. You ask your friend the question and your friend says "I don't know. Let's inquire." That means "let's go inside the store and ask the question".

This answer is for British English (I think American English doesn't use "enquiry" spelt with an "e"). Both words have the same basic meaning but "enquiry" is more the process of asking a question whereas "question" is just the question itself. So in class you could say, "I have a question" or the teacher could ask, "Any questions?" but you wouldn't use "enquiry" instead. You could write, "Have there been any enquiries about the job we advertised" but you couldn't use "question" in this sentence. Both can also be used as verbs but again in different ways - for example, "I have to question your use of that word" (meaning I have to challenge it - enquiry cannot be used to replace question in this sentence). A second example: "I would like to enquire about the cost of renting this apartment" (question cannot replace enquire in this sentence). Note also that in British English "Inquiry" is usually something formal, like a judicial inquiry (a formal process of investigation). I think American English uses "inquiry" for both meanings and increasingly so does British English.

A question is a sentence ending with a question mark (?) An enquiry is asking about something. It could be as simple as a question, or it could be more. An inquiry is a formal investigation.