I don't know where you want to start or finish, but here's some strava routes for central Durham to to central Carrboro
Red roads are more popular for cyclists, blue ones are less travelled, and white ones are rarely or never cycled.
Route between Durham, NC and Carrboro, NC? This one is the most "popular" based on strava tracks, and is 22.8 km.
This path minimises elevation changes, so instead of 309 metres of vertical you only have 225 metres of climbs.
This is the most direct path but it appears to go down a main road, which may not be legal on a bike, in your area.
You can also find a number of strava segments in the area, using this link
Curiously, Google's suggestion is different again to any of these.