It’s a great question. The answer is yes. Think of the big meteoroid that hit the earth 65m years ago and extinguished the dinosaurs. 10km in diameter and trillion tons of weight. That must have created billions of tons worth of rocks flying off earth into space. Scientists have calculated where the material could have ended up. So one result I saw is that Europa, one of the moons around Jupiter, must have gotten hit by rocks from the earth as much as our very own moon. So think about it. It’s not just rocks, its water, life, that went into space. What I don’t know is if we have found remnants of these rocks in human moon expeditions.
I think you mean rocks from mars and rocks from earth and I agree with Kamal’s answer.
The term asteroid has a specific meaning. An asteroid is a rocky object orbiting the sun. Think of it as a small planet! The asteroid belt lies between the Earth and Mars. Mars does not generate asteroids as your question suggests.