The Questure are part of the State Police of Italy they deal with all immigration issues, the issuing or permits to stay and passports, to everyone whether you are Italian or not. In some parts of the country some things the Questure do are dealt with by the Prefecture in the city or province.
Each region and province has their own website, so you will need to go directly to your local site for information.
The present organization of Public Security in Italy was introduced by Law N. 121 of 1981. The National Authority for Public Security is the Minister of the Interior, responsible for public order and security, and the coordination of police forces. In Italy there are five police forces: Polizia di Stato, Arma dei Carabinieri, Guardia di Finanza, Polizia Penitenziaria and Corpo Forestale dello Stato.
The Public Security System in Italy
Enforcement of public order and security policies is entrusted to the Department of Public Security, headed by the Chief of Police – Director General of Public Security. The Chief of Police is designated by the Minister of the Interior and appointed by the President of the Italian Republic, subject to the resolution of the Council of ministers. He is assisted by a First Deputy Director General, a Deputy Director General responsible for the Co-ordination of Police Forces, and a Deputy Director General in charge of the Criminal Police Central Directorate.
The Department of Public Security coordinates police forces’ operations, and manages and organizes the Polizia di Stato – Italian National Police. The Department of Public Security consists of a Secretariat, 13 Central Directorates and 4 Offices. Some of them have an interagency status as they are staffed with personnel from the Polizia di Stato, but also Guardia di Finanza and Carabinieri.
They have a good page in English, although you can select other languages explaining a lot about The Questure and also a specific section aimed at foreign nationals.