I will suggest you different places. You may not find them useful but they work pretty good. You could go at first in countries like Australia, France, Greece, Canada or Switzerland that are really pet friendly. Of course you need to have a pet passport but these are countries that will give you amazing memories.

Another place this time in the US would be Cumberland Falls in Kentucky with some of the most amazing waterfall scenes

Lastly if you wanna stay close to Georgia then i would clearly suggest you Desoto falls since the location is wonderful and it looks really peaceful.

Hope you found it useful :)