Coming to Qatar just to learn Arabic may be a problem since about(just an estimation) 80% of the population here are non-arabic speaking. About 20% of the people here are actually Qatari. And Qatari people speak English when communicating with the rest of the population.

If you enroll your children in a school here then I am sure that is the best way to learn Arabic. But enrolling children in schools are not easy....considering money, location,spot available for your children, and subject emphasis as factors. And of course television, and interacting with people who speak Arabic is another way to learn. But I found that people of the same kinda stick with each other which is unfortunate.

Hope this helps. Good luck!