Answer:
True.
Step-by-step explanation:
In July 1861, the Confederate general John R. Baylor won the war with Federals over the southern parts of New Mexico and Arizona.
Since then between 1861 to 1862, the name of these parts were renamed as Confederate Territory of Arizona.
The New Mexico territory was the remote area of America, on which both Federal and Confederate claimed ownership. But in July 1861, Confederate Colonel John R. Baylor claimed its ownership and renamed the territory as Confederate Arizona.
So, the answer is true.