Black slavery was never a significant institution in the New Mexico Territory (which included modern-day Arizona, New Mexico, and southern Nevada) because a) there were other institutionalized forms of servitude, b) slaveholders settled in the territory just before the Civil War broke out, and c) there was an absence of any large scale agriculture (which would require vast quantities of slave labor).

In regards to a), white New Mexicans were in near-constant conflict with Native American tribes like the Apache and Navajo (particularly in the south). While skirmishes with Amerindians were initially challenging to white settlers, they soon began to take advantage of the hostilities by invading Amerindian domains and capturing Navajo and Apache families. The settlers forcibly employed these captured Amerindians into "peonage," an indentured servant-esque system in which the Amerindians would work to pay off debts imposed on them by the whites. The settlers would pay the Amerindian servants paltry sums for them to work on their ranch estates (which the Amerindians could theoretically use to pay off the debts they incurred). However, there was a catch: the settlers would make the Amerindians pay exorbitant prices for food, lodging, and tools, forcing these workers into even more debt and ensuring that they would never be able to escape peonage. Basically, peonage was just another form of slavery (though unlike chattel slavery, the Amerindians could not be bought, sold, or separated from their families), so there really was no need for more slaves. In fact, the only slaves brought in were blacks who served as body slaves (read: Gary from Veep but unpaid) and accompanied their owners to New Mexico. If you're looking for a specific number of black slaves, this article says that New Mexico never had more than a dozen.

As for b), you need to keep in mind that New Mexico was a relatively new territory that the U.S. acquired only after the Mexican-American War, which ended in 1848. Tucson and Masilla, two towns which contained the largest populations of white who supported the southern cause, weren't in American possession until the Gadsden Purchase of 1853. There just wasn't really an opportunity for slaveholders to establish strong roots in places like modern-day Arizona.

Also, there just weren't that many people living in the New Mexico Territory in general. When slaveholding whites wanted to split off from New Mexico and form the new territory of Arizona (which consisted of modern-day southern New Mexico and southern Arizona*), they appealed to the Union but were rejected (because no government official believed that they had 8,000-10,000 settlers).

In addition, the people who settled in the southern parts of the territory and supported Southern black slaveholders weren't particularly concerned about the institution itself; rather, they just wanted to get on the good side of southern lawmakers because they wanted these lawmakers to build a transcontinental railroad through the territory of the Gadsden Purchase. They thought a railroad would boost commerce, and because they didn't really care about the plight of non-whites (as evidenced by peonage), backing slavery was an easy concession for them to make.

*Modern Arizona is very different from the proposed Confederate Arizona, as it is comprised of the western half of the former New Mexico territory, whereas Confederate Arizona was comprised of the southern half of Arizona.

As for c), there was simply no need for large quantities of slaves. As mentioned in b), New Mexico Territory was only recently acquired, so there wasn't a chance for plantations to be established. Also, as implied in a), the Arizona and New Mexico territories were basically still frontierland, and there would often be fighting between Amerindians and the white aristocracy. It's simply infeasible for settlers to build a plantation if the fields are at risk of being raided every other day. That's why the largest landholders in the territory were ranchers, who would be able to transport their livestock quickly, minimizing losses to Amerindians.

So, basically, there was some slavery in Arizona, but the vast majority of it was not black slavery.

Sources:

El Gringo: Or, New Mexico and Her People (W. W. H. Davis) --> written in 1857, lots of interesting details about peonage and the treatment of natives in the New Mexico Territory

"SLAVERY MUST DIE": Radical Republicans and the Creation of Arizona Territory (Deren Earl Kellogg) --> goes into detail about the politics of the Arizona territory (also the source where I learned about the true intentions of slavery supporters in Arizona)