I am not sure if roundworms are as species specific as other parasites. It certainly wouldn't hurt to take a fecal from your dragon to the vet for testing. To avoid cross contamination, deffinitely wash your hands throughoutly before handling your dragon, food dishes, prey items and salads. Don't let your dragon run around anywhere your cat commonly lies or where her litterbox is. Probably be best to keep your cat away from your dragon completely.
Bearded dragons can get several kinds of worms and parasites, including round worms. They can also have hookworms, tapeworms, coccidia (and other protozoans), pinworms, flagellates and others.