Doha does have some music classes and other similar "developmental" classes. Gympanzee (in the Villagio) is a good place for them. The website called The Word (http://www.thewordinqatar.com) has some "structured playgroups" listed.

You might also want to check out Doha Expat Mums & Kids. We offer both structured and unstructured playgroups, in addition to many other activities. We're a bit slow right now because so many expats have left for the summer (myself included) but it will pick back up in September. http://www.dohaexpatmumsandkids.com