That is not accurate. The child's surname has absolutely nothing to do with child support obligations, and courts can allow a name change over the father's objection.

If mom wants to change the child's name, though, she either needs the father's consent for the name change or she has to initiate court proceedings and convince a court that the name change is in the best interest of the child.

Maricopa county puts forms and instructions for name change petitions online, with the instructions covering both situations where the other parent consents and where he objects; many other counties likely do, as well.