There are Many places that offer Arabic Language courses as it is one of the main requirements by Expats in this country. You can see a list of places listed on this website:

http://www.qatarvisitor.com/index.php?cID=414&pID=1077

You can also find people who are native speakers and would be willing to assist you depending on what you want to learn exactly. I would be one of them.

Hope the above is Useful.. Good Luck!