Yes you can. You sponsor has to go to Immigration and ask for exit permit for you, because the people who come with family visit visa they can go out Qatar and back again during the validity of the visa. I know some people the bring their wives with family visit visa and and make for them Exit permit then they can travel to Dubai or any country and back to Qatar without any problem. I asked personally one person who is an employee working in the Immigration about that case and he replied that no problem according to the aforementioned.