Both 'query' and 'enquiry' (inquiry) mean a request for information. 'Query' is also used to mean a question. 'I have a query/ question about my order
But 'inquiry' ('enquiry' is a British spelling) can also mean either a) an official effort to collect and examine information about something. 'The police are conducting an inquiry into the circumstances of his death.' or b) the act of asking questions in order to gather or collect information. 'Further inquiry showed that he had visited the city twice before.'