Spire Rock is a small Forest Service campground about 15 miles southeast of Gallatin Gateway.
Other good bets are Moose Creek Flat, five miles north of Big Sky [1] on Highway 191; Swan Creek, just north of Moose Creek Flat; and Greek Creek, just north of Swan Creek.
On Highway 191 about eight miles south of Big Sky, Red Cliff is a riverside Forest Service campground about 50 yards off the highway.
These campgrounds are part of the Bozeman Ranger District of the Gallatin National Forest (3710 Fallon St., Bozeman, 406/522-2520, www.fs.fed.us/r1/gallatin [2], mid-May–mid-Sept., $11) and can be reserved online (www.recreation.gov [3]).
Links:
[1] http://www.moon.com/destinations/montana/the-missouri-headwaters-and-south-central-montana/big-sky-and-the-gallatin-valley/s/big-sky
[2] http://www.fs.fed.us/r1/gallatin
[3] http://www.recreation.gov