I'm going salmon fishing on the South Fork of the Salmon River. Since the limit is only one fish per day, can I catch and release salmon after I keep one?

No. Once you attain a daily bag, possession, or season limit you must stop fishing immediately. By the way, those rules apply statewide. So if you catch and keep a salmon on the South Fork you can't go to the Little Salmon and catch another, since the daily limit there is also one fish.