Apple: Your toilet tank is desperately trying to get your attention.
If it could speak it would tell you that it is need of a new flapper. The one that is in there now is allowing water to slowly seep down into the bowl. It could also be that the refill tube is inserted to far down the overflow tube and is slowly siphonig water into the bowl.
When the water in the tank reaches a critical level and you open the faucet, the reduction in line pressure will induce the ballcock to refill the tank.
Why not have a party for your toilet tank and replace the ballcock too? Be sure to use an anti-siphon ballcock.
Lots of luck…Akmed