Barnton is likely to be the village located in Cheshire in England. Its Wikipedia page mentions their football team, but does not reference an amateur Quidditch team (Wikipedia).
There is an affluent suburb of Edinburgh in Scotland called Barnton as well, but the Cheshire village seems more likely as the historical location for a Quidditch team.
Lexicon timeline of Quidditch
Lexicon list of British and Irish Quidditch Teams
From the Web
Barnton Village Council: Barnton Village
Harry Potter Wiki: Barnton
WizardingWorld (Pottermore) features: