After flushing the toilet and the tank starts to refill, just as the tank is full a noise occurs.

The fill valve has a diaphram or washer that makes the seal, it is loose and vibrating. What type of valve do you have? https://www.askmehelpdesk.com/attach...20ballcock.gif or https://www.askmehelpdesk.com/attach...-20regular.jpg

My bet's on a float ball type of ballcock, (see image).
Turn the water off and remove the three screws on top and pull the float ball and stem assembly out. Pry the washer out and reverse it. Seat it in firmly. Now hold your hand over the open ballcock and turn the water on for a few seconds to flush out the supply. Reassemble and give it a flush. Better now? Tom