In general I think both places are pretty safe, and have pretty large tourist industries especially UAE/Dubai.

Are you planning on staying at tourist areas/resorts or traveling around on your own? Either way make sure you do a little research and obey the local laws/customs. Both are quite conservative countries- if you're at a resort hotel or something they're much more lax but outside those areas you need to pay close attention to stuff like modest dress, public displays of affection (I think thats illegal in Qatar), etc.