No, I think all you need to do is clean the area of that fill valve near where the rubber diaphragm is.
What you need to do is this:
1. Shut off the water to the toilet
2. Remove the three or four screws holding the top on the fill valve. The little button that the float arm pushes down on is connected to the rubber diaphragm, and you want to clean both sides of the rubber diaphragm and the area of the fill valve above and below the rubber diaphragm.
3. Once you remove the screws holding the top on that fill valve, you should be able to remove the top, float arm and float as an assembly. You should now see the rubber diaphragm at the top of the fill valve.
4. Remove the rubber diaphragm, and look to see if there was any dirt or anything under it. If the rubber is rotting so that it leaves black stains on your fingers, it means you need a new rubber diaphragm. If you can find a manufacturer's name on your fill valve, you should be able to buy a replacement at your local home center or hardware store.
5. After cleaning and/or replacing the rubber diaphragm, put the fill valve back together and turn your water back on. Problem should be fixed now.
Most likely either the rubber diaphragm is rotting and needs to be replaced, or some dirt just got caught between the diaphragm and the seat it seals against.