According to Google.. That would be a 'Neighbourhood' in English :)

'District' is a possible translation. 'Neighbourhood' (GB) or ''Neighborhood' (US) would also work. This is used more in US English. The word 'quarter' does exist, but it has quite a specific use, such as the 'Latin Quarter' of Paris, or the artist's quarter of some cities. I wouldn't use it to translate 'quartiere'. Often, though, we just use the more general term 'part' or 'area' e.g. There is a lot of crime in the area of the city where we live.