M.B.
Basically, it is a vibration set up by flushing the toilet that is reverberating throughout the house.
One thing you can do is to reduce the amount of water that flows into the toilet... just turn the handle behind the toilet a bit, flush the toilet and see if that fixes it.
If not, turn it a bit again, and try again. Keep trying until that stops the noise. Hopefully you will find that point where the noise stops, but water will still come in.
Sometimes this helps... it will take longer for the toilet to fill, but it will still put the same amount of water into the tank, and reduce that vibration.