The square kiva located inside the convento of the Quarai Mission is an interesting riddle. What is a subterranean Puebloan religious structure doing within a Spanish mission? And why is a regularly circular kiva in this case square. Just like the kiva inside the convento at Abo, we do not really have a good answer for these questions.
What we do know is that this square kiva was built sometime between 1622 and 1645. This would have been at the same time that the Quarai convento and church were being constructed. The placement of this kiva exactly in the center of the larger patio room suggests that this was not a kiva that existed prior to the construction of the convento, but rather was built in this location on purpose. What this may suggests is that the square kiva was built under the supervision of the Spanish missionaries possibly as a conversion technique. The Spanish recognized that in conversion you would be less successful if you simply disregarded the previous religion.
As for the shape of the kiva, the square design is actually common amongst the western Pueblo groups. This would include the Hopi and Zuni, two tribes that trace their ancestry to the Salinas Pueblo Missions.