Quarter sessions were held four times a year and were also presided over by Justices of the Peace. These records for Petty Sessions and Quarter Sessions are usually held at the Local Archives.
I would look at newspapers, trade directories and street directories if you know where the person lived.
I don't know if any Oaths have survived and where they would be kept. I know they are available for Scotland from National Records of Scotland. If they are not held at the local Archives in Court records then it might be worth checking on National Archives site, search Discovery or look at the research help pages. Maybe the local Court still retains some records.
I've found a couple of references in newspapers of someone being appointed as a JP. JP's names are usually given in newspapers for the reports on Petty Sessions, Police Courts, Quarter Sessions etc.
Try searching the local Archives online catalogue, use person's name, Justice of the Peace and also petty sessions, quarter sessions in your search.