OK a clue is found
Try looking to the left bottom side of your toilet tank for a valve.
Close this valve slightly and flush again.
Chances are it is either a bad angle stop (valve) supplying the toilet OR a bad fill valve ball cock.
The easiest one to replace is the ball cock.
A 400A is code approved as it has a built in vacuum breaker and the entire device can be bought for about $4.00
The directions are very easy to follow for the installation and adjustments and takes about 20 minutes.
My normal installation charge is around $125 + BUT every other place in the country has their own pricing. This may give you a budget idea
The hard part is finding the exact source of the noise and BY golly you did it hopefully. Good luck and if you have any questions feel free to E mail me