We know that they had martial arts. Apart from it being strongly implied, even necessarily so, by the existence of skilled professional warriors, we know that these skills were systematically taught. For example, Bernardino de Sahagún noted that military skills (along with other skills and knowledge, including law, theology, astronomy/astrology, etc.) were taught in the calmecac schools.

We don't know the details, but pre-Columbian martial arts would have included the use of the major weapons. They would probably have included wrestling.

The modern Peruvian martial art of Rumi Maki claims old roots, and the part that teaches skills with the old weapons (sling, spear, bow, spearthrower, mace, axe, club, and knives) might preserve pre-Columbian skills.

We have much more detail on later - post-Columbian, but still indigenous - martial arts in North America, due to the plentiful writings on indigenous culture and fighting, based on first-hand experience and observation, into the 20th century. For example, Charles Eastman, who had a traditional Santee Dakota childhood, wrote

Wrestling was largely indulged in by us all. It may seem odd,, but wrestling was done by a great many boys at once - from ten to any number on a side. It was really a battle, in which each one chose his opponent. The rule was that if a boy sat down, he was let alone, but as long as he remained standing within the field, he was open to an attack. No one struck with the hand, but all manner of tripping with legs and feet and butting with the knees was allowed.

(C. Eastman, Memories of an Indian Boyhood, McClure, Philips, 1902)

Many of the martial skills described in that literature are obviously post-Columbian (horsemanship, mounted archery, musketry), and some are likely to be continuations of pre-Columbian traditions (wrestling).

Apart from Rumi Maki, there are other American martial arts that claim to have an indigenous component (but not always pre-Columbian). Typically, the only evidence supporting such claims is the internal oral history of the martial art. The place to look for authentic traditions is such arts is weapons. Unarmed components (wrestling and striking) may have existed in the original martial arts, but, other than sportive wrestling, would have been only minor components of a weapons-oriented tradition. In modern arts with claimed indigenous roots, the unarmed component is often the primary component, and cannot be trusted to have preserved pre-Columbian techniques (or even any particular old, but post-Columbian techniques). Even weapons techniques in such arts can be modern re-constructions of lost skills.

Reference:

J. Ramon et al., Rumi Maki Fighting Arts, Blue Snake Books, 2007.