Yes, one parent can relocate with a child to another city, county, or state. However, they should always seek court approval prior to doing so.
When there is a court case that is already within the North Carolina system that lays out child custody, relocation without first seeking court approval can lead to very messy circumstances. Oftentimes, relocating without court approval can result in not being granted relocation approval at a later date. This means you would have to move the minor child back, per the original custody arrangement.