From my experience, you would require an exit permit for your mother, given she'd stayed for more than 1 month on visit visa. However, you don't have to worry, there is a small counter behind 7 or 8th line at Hamad Airport, who collect penalties and fines, etc. Show them ur QID (assuming u r the sponsor), they will process it and will give u the print out. Give it to ur mother to be shown at immigration counter while leaving for boarding. That's it.