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Sorry to hear about your problem.
This noise is typically the fill valve located in the tank. The newer fill valves, as shown below, use a float rather than the old plastic ball on the end of a rod. If you have this type of float valve, you most likely can just flush any sediment that has managed to get caught in it. Depending on the brand, some twist off near the base, some have a cap on top that twists off after lifting off a plastic cover. With the water supply shut off, located under the tank, you can twist off the fill valve and then turn the water supply back. Just make sure you use your hand or a cup to block the flow of water that will come out of where you just untwisted the fill valve. This will flush out the sediment. You can also clean the rubber washer that is located here. put everything back together and it should work fine. If you would rather replace the fill valve, you can do this is as well since they are inexpensive.
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Best of Luck, Brian
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Well this make it a little harder to diagnose, since I cannot see it firsthand. I would verify that the tank has absolutely stopped filling, because this is a typical noise that occurs especially when the fill valve is completing the fill.
If there is no supply water moving at the time of the noise, then I would have to guess that it is in the drain piping. I would look for any areas where the piping may be rubbing against something, as the water drains. If you have a basement or crawlspace, have someone down there to listen while the toilet gets flushed.
Once the toilet fills and the fill valve shuts off, this sudden stop can cause vibrations to be sent back thru the supply lines. This can cause water hammer, in which a water hammer arrester would help to fix the problem. The supply piping could also be loose and hitting something nearby. It would be good to verify that all piping is adequately secured to the framing. A water hammer arrester is shown below; it reduces the shocwave sent out by the sudden stop of water.