The terms enquiry and inquiry are considered to be different spellings of the same word. Both terms are nouns used to refer to “an act of asking for information” or “an official investigation.”
In its traditional sense, enquiry and its corresponding verb form enquire are used for the general sense “ask.”
“Enquiry replies are key element when you are selling any property”
“Manchester United transfer news: Mourinho makes enquiry for £25m international to solve defensive headache”
“Women are engrossed in TV soaps, forget to enquire about husbands: Goa minister”
“Zehri Visits CHK to Enquire after Noorani Blast Injured”
On the other hand, the term inquiry and its corresponding verb form inquire are used for formal investigations.
“Trump’s Taiwan phone call preceded by hotel development inquiry”
“Matt Barnes of Sacramento Kings Sought in Assault Inquiry”
New York Times
“Aurora police chief: We are not going to inquire about immigration status”
“Panel set up to inquire into sexual harassment allegation against CEO Rakesh Sarna: IHCL”
However, in current practice, writers do not really distinguish between these words’ traditional uses. It is more common to see the terms enquiry and inquire in British English publications while American writers tend to use inquiry and inquire.