My son lives with his mom but wants to move in with me do I still have to pay her child support we both live in Caifornia

First your son can not just move, because he wants to, if he wants to, and it is OK with his mom and you ( you both agree) then you both file for a motion to modify custody and support.
You get custody and she would start paying support according to her income

You have to change the custody order, or you will have to keep paying