Yes — many hickory nuts are edible, but taste varies a lot by species. According to the Missouri Department of Conservation, shagbark hickory (Carya ovata) produces large, sweet, relatively easy-to-crack nuts, and shellbark hickory (Carya lacinosa) is noted for some of the best-tasting hickory nuts in Missouri. Other species, such as bitternut and water hickory, are bitter and are mainly wildlife food.
Ripe nuts usually drop in September–October. As with any wild food, identify the tree carefully, discard moldy or rancid kernels, and do not eat nuts you cannot confidently identify.
Source: Missouri Department of Conservation — Hickories field guide