Yes, that's possible, and it's actually pretty easy. Just relax your tongue and throat, open your mouth and breath. Most air will go through your mouth, but some of the air will go through your nose. You can make a bigger portion of the air go through your nose by moving your tongue to restrict air from flowing through your mouth.
However, I don't think it's possible to simultaneously inhale through your nose and exhale through your mouth, or vice versa.