On God is a phrase which is a way of saying that someone is telling the truth. Similar to another phrase such as "I swear to God." Other common uses are "On my mom," "On my grandma," "On my grandpa," "On gang," etc...
Person 1: I whipped out of that parking lot so fast, spinning the wheel, I 'bout flipped the damn car over. Then the police started following me, and I had to run away.
Person 2: Really?
Person 1: On God, bruh."