The noun enquiry is a British spelling of the word "inquiry." Both words mean the act of asking questions to gain information. If you are in London, you can make enquiries about the opening times of the many museums there.
Although enquiry is now considered the variant spelling in the United States, it was the original English spelling of the word that comes from the Old French word enquerre. The noun also means a systematic investigation, usually undertaken for the benefit of the public. If you discover pollution in your local river, the government may open an enquiry into the source of the pollution and ways to clean it up.