A woman is claim that her son is mine but is refusing a DNA test. How can I get a court order to get it done?
You can file a Complaint in Paternity, not necessarily claiming that you are the Father, but asserting that she has alleged that you are, and asking for paternity test. That request would be easily granted.
Per A.R.S. §5-807, “The court, on its own motion or on motion of any party to the proceedings, shall order the mother, her child or children and the alleged father to submit to genetic testing and shall direct that inherited characteristics to determine parentage, including blood and tissue type, be determined by appropriate testing procedures conducted by an accredited laboratory. If the mother is unavailable or fails to cooperate by refusing to submit to genetic testing, testing of the alleged father and child or children may be appropriate. An expert duly qualified as an examiner of genetic markers shall be agreed on by the parties or appointed by the court to analyze and interpret the results and report to the court.”
The court can be asked to Order you both to share the costs of the tests.