I'm a very constipated person, is it normal to only pass gas when having a bowel movement?

If you are lucky. Unless you are planning to ruin the morning elevator commute, you are pretty lucky if that's the only time. You could be the person that toots every time they take a step. The more important thing is to work on that constipation in order to deter the long term effects such as diverticulosis and hemorrhoids.