Yes, of course, you are able to do that, but Qatar makes it kind of difficult (so what's new?).

Basically as a single mom, to sponsor a nanny/domestic worker you have to provide a copy of your marriage certificate or your divorce certificate -- something to show that at some point in time you were married.

What they would do if you weren't ever married, I don't know. It's not illegal in many other countries to be a single, unwed mother.
Or what if a single woman wanted to adopt a baby? What then?

They've made no formal provision for these situations, I think.