Inca is not a river. The Inca Empire (1438–1533) was the largest pre-Columbian civilization in the Americas, centered in the Andes of modern-day Peru.
Major rivers within the empire included the Marañón, Ucayali, Urubamba (flows past Machu Picchu), Apurímac, and Titicaca basin—but none are named "Inca."
"Inca" appears in species names (Inca tern, Inca dove) or brands. The empire's name derives from Sapa Inca ("unique ruler").