Yes, but for best results, make sure you have the kind of glass paints you can heat set in a home oven. Pebeo is one example that usually costs under five dollars each. Start with a clean ceramic mug and either stencil or brush on your design with the paint. Follow the manufacturer's instructions for baking time and temperature.
Delta makes an awesome glass paint that doesn't require heat curing. It is important to condition the surface first, so that paint will adhere properly.
You can also use foam stamps and apply the paint to the stamp. Then stamp this paint applied foam stamp onto the mug which needs to be painted.
There are glass and ceramic paint. Check your local craft store. To prepare the mug, wash throughly and wipe with a alcohol wipe. Let dry throughly. Then its ready to paint. Do not paint the inside of the mug or the area that your mouth touches. Let dry 24 hrs. Do not wash in dishwasher.
I am assuming you're using glass paints interchangeably with china paints in this case, and while you probably could use them on an average cup I would recommend against it. The glazes that are used in commercial stoneware often are not set up the same as the ones we would use in conventional ceramics and may interact badly