Not within the park. However, this is Arizona so there are plenty nearby. I recommend the hiking, biking and running trails in GR if you are capable.

Off road for ATVs? I’m not aware of any. Lots of hiking trails however!

Tucson Mountain Park which surrounds Gilbert Ray and the Saguaro National Park has extensive hiking and mountain biking paths - a wonderful system. But to the best of my knowledge, that area doesn't have OHV trails. I found this link which might help - https://www.alltrails.com/us/arizona/tucson/off-road-driving