Can my mother take my child if I move out-of-state?

Unless your mother can prove that you have abused or neglected your child, I think this is an empty threat.  As long as your child is healthy and well cared-for, and you can support yourself and your child in your new home, there's probably nothing to worry about.  If you have mental health or substance abuse issues, however, that have been documented already, that might be a problem.

Typically, grandparents don't have a lot of rights to custody or even visitation, against the parent who is their own child.  And courts don't like custody changes, because they are very hard on children even when they are necessary, so ordinarily it will take a pretty clear case to get a court to act.