When you trade your car does it matter how much payments you still have on your car?

The number of payments makes no difference. The only thing that matters is the payoff balance of the car vs. the actual cash value. The more payments you have left the more likely you will owe more on the car than it is worth in which case you would need to either come up with more down payment or roll any excess balance onto the new loan which typically requires better credit to do.