My upstairs toilet sounds like it is "trickling" or "bubbling down the wall but no water damage. I just replaced the fill valve and ensure the water stops appox. 1" below the overflow pipe. If I turn off the water supply the noise stops and if I turn the water back on the noise remains gone, however, if I then flush the toilet the noise returns. Any ideas?

Sounds like a worn flapper valve or possibly a worn Douglas valve.

I agree, a worn or not fully seating flush valve will allow water to trickle into the bowl from the tank, this causes water to trickle out of the bowl down the sewer pipe. Could just be a bad flapper which is cheap and easy to replace.

Several things come to mind, Well, three to be exact.
(1) Run you finger around the seat. There should be no rough spots or nicks. If there are replace the entire flush valve,(seat and overflow tube).
(2) The flapper chain must have 1/4" of play when the flapper's seated and the tank filled.
(3) Flappers come with a ring that you slip over the white overflow tube to install on older flush valves,(the seat).
Newer tanks have small hooks at the base of the white overflow tube for the flapper to hook into. If your tank has those hooks and you didn't cut the ring off on the lines indicated with a pair of scissors and install it correctly it will seep water from the tank into the bowl. One of the above should solve the problem. Hope this helps and thank you for rating mt reply. Tom