No
Bitcoin Gold has replay protection. After the fork, all further transactions on the BTC chain will not transfer BTG, nor vice versa.
In your example, when someone sends you 1 BTC after the fork, the corresponding 1 BTG will still belong to the sender, not to you. You will not gain any BTG from the transaction. (Of course, they could send it to you separately if they wanted.)
It will also not be possible to spend BTC as BTG or vice versa. To exchange one for the other, you will have to find someone to make a trade with, and agree on an exchange rate, which will probably not be 1:1. (You can use an organized exchange to make this efficient, but it's the same thing in principle.)