County performed well against Celtic last season, taking five points from their three home games with Neil Lennon's side, but they were soundly thrashed 4-0 on their only away trip.
Their first home game comes seven days later when newly-promoted Partick Thistle will be the visitors to Victoria Park.
County will be looking to emulate their success of last season, when they finished fifth just behind local rivals Inverness, who they meet for the first time at home on October 26.
Boxing Day will also be spent at home, with Hibernian the visitors, and after a December 29 clash at Aberdeen, they make their only trip to Inverness on New Year's Day.
The season splits on April 5 and County's last fixture is a third derby clash, this time at home to Caley Thistle.