From Wikipedia: "In Standard Arabic, the name is pronounced [ˈqɑtˤɑr], while in the local dialect it is [ˈɡitˤar]." I'm not so good with the International Phoenic Alphabet. What does this mean in common parlance?
In standard Arabic, Qatar is pronouned with the first syllable stressed and a sound that doesn't exist in English - sort of a K sound in the far-back of your throat made by clicking your tongue against your uvula. That sound has dropped out of a lot of modern colloquial Arabic dialects and has been replaced by other sounds. In the Gulf states and in most Bedouin communities, it's been replaced with the hard G sound, just like in "go." So the closest pronunciation you could get to how Qatar is said colloquially in Qatar would probably be something like "Guitar" with the first syllable stressed instead of the second syllable. That being said, basically any of these are valid pronunciations in some dialect, just given how varied Arabic is and how imprecise the art of transferring these sounds into English is.
This video has several examples of how Qatar is pronounced in the standard/classical Arabic style starting at around 0:44
https://www.youtube.com/watch?v=9MnaLC2BRw4I couldn't find any videos of people saying it in colloquial Gulf Arabic, mostly because people don't tend to record themselves having casual conversations about countries and posting them on YouTube.