Here is an option with a couple short hikes in the area:
Petroglyph and Arch Canyon
These two canyons are located just north of I-70, on the east edge of the San Rafael Swell. From the town of Green River, drive west on I-70 for 12 1/2 miles. Just west of mile marker 147 you'll see a graded road that heads north from the Interstate. Drive that road for .7 miles (closing the gate behind you). At the .7 mile point you'll see a track forking west (left) which leads to the start of the Pictograph Canyon hike. Travel down the track 1/4 mile and park under the cottonwood trees.
There are several decent campsites near the trailhead.
About .6 miles from the parking area the trail drops down to the bottom of the wash, near the junction of Petroglyph and Arch. At the junction, Petroglyph is on the left and Arch Canyon is on the right. Both canyons are short and easy to hike. It doesn't matter which you do first.