Yes—though not "board games" in the modern Eurogame sense. They played games of skill and chance on marked surfaces:
The most famous. A cross-shaped board with 52 squares (representing the 52-year calendar cycle). Players moved 6 jade/bean pieces per turn based on throws of 5 marked beans (dice). Combined racing, gambling, and religious ritual. Played by nobles and commoners; high stakes included freedom or property.
Simpler: toss a gold bean onto a square-marked mat/target. Gambling game, less strategic.
Not a board game, but a ritual sport on a stone tlachtli court. Heavy rubber ball, hip-only contact. Deep cosmological significance; sometimes culminated in sacrifice.
Patolli is the closest analog to a "board game" and was culturally central.