You only get the extra gold if those tiles are being worked by your citizens. If you want to manually assign them to the tile, zoom to your city and click the gold tile. However, this means that another tile is now not worked, and that'll usually mean you lose shield or food production because mountains aren't very productive. It's up to you which you need more, gold, food or shields.
It's hard to be specifc, but a good way to improve your gold situation is switch to republic and build roads on every worked tile.