I figured we had a couple of potential candidates causing the sound. You seemed pretty sure it was related to the hot water being used and the starting of the water heater. ("every time the hot water is used and the water heater turns on the fog horn noise happens.") So I figured it must have something to do with the hot water flowing or the only other thing that would be flowing, the gas.
I have no experience or reason to believe it would be the gas but only that it was a candidate. Since we know sounds such as these are typically made by passing a liquid or gas through an orifice, and possibly associated with something that's loose which could amplify the sound. So if was happened when the heater started, without water flowing, we could, I believe, eliminate the water as the possible cause. And if the sound was just at start-up doubtful water expansion would be involved.
I suppose there still is the exhaust flow that would be tied in with the firing up of the heater.
So you say, "The noise does start when the gas comes on initially" but didn't mention if that was without the water flowing. ???