I am looking to learn Arabic and have found a wide variety of different versions. I wanted to know what version of Arabic is most commonly spoken in Qatar?
The variant of Arabic you are looking for appears to be Gulf Arabic, which is spoken "around the coasts of the Persian Gulf in Kuwait, Iraq, Bahrain, eastern Saudi Arabia, the United Arab Emirates, Qatar, Iran and northern Oman" (quoted from Wikipedia; emphasis added). There are regional variations in Gulf Arabic, though, so for any course or resource for Gulf Arabic, you will need to bear in mind that the variant spoken in Qatar will have some differences.
The website GulfArabic.com provides a number of free lessons, but for full access you will need to pay. The site claims that it offers "the #1 Gulf Arabic comprehensive e-learning course"; it provides "audio for every Arabic word or sentence".
Another resource is Colloquial Arabic of the Gulf by Clive Holes (Routledge, 2nd edition, 2009). This is a beginner's course; I couldn't find any audio materials that accompany the book.
There are also a number of older books by Hamdi A Qafisheh, including the following: