The short answer is NO, you will not be able to run 31's on a completely stock sidekick without major rubbing and possible tire failure due to the fender wells cutting into the tire itself.
Also, you would want to learn more about the rim, particularly what 5 lug dimension these are. There are several 5 lug configurations, and only 5 lug with a 5 1/2" lug pattern will work on a Sidekick. If these are 5 on 4.5", they are more than likely meant for a Jeep TJ, etc. If you ask them what they came off of, you can probably deduce whether they would fit or not. Vehicles like older Broncos, and Jeep CJ's use the same 5x5 1/2 pattern that we use.
If you are considering 31" tires, there are several ways in which to accomplish this. Budget, vehicle condition, and mechanical ability all determine what approach you would take to lift your vehicle.
Hopefully this helps a bit. If you want/need help on lifts, all of us on here can help you determine what would be the best approach.
-Zukisidekick