my understanding is that Qatar is not the correct pronunciation and Gatar is closer (though not perfect about 90%) and as for the Queen being a Gueen, it won't work. The U is what makes Queen sound like kween, as Qatar has no U it sounds more like Katar than Kwatar.

The Queen's advisors would have checked with the Qatari Embassy to make sure it was as correct as possible for a non Arabic speaker.

before you argue your profile says "I am not a Qatari" yet your Nationality says "Qatar" So you need to sort out which you are before criticising the Queen for her verified pronunciation.