If I take my mom to family medicine clinic, can she pick up a childs germs?

Risk of Infection. Yes theoretically it is possible but in practice you need a close contact with children to get infected with their germs. The risk is more in a child to child exposure as you get older you get immunity to many common childhood infections. As long as your mum doen not come close contact the risk is very minimal as long as there is appropriate seating in the waiting room and the waiting period short.
Of course. Wherever there are children, there is a risk of getting exposed to an infectious disease, just like going to the mall or going to a movie. Best protection is basic hygiene, washing hands and coughing into a hankie or napkin.