Yes, it is true that the Colorado River flows through the Grand Canyon. Rising in the central Rocky Mountains in the U.S., the Colorado River flows generally southwest across the Colorado Plateau and through the Grand Canyon before reaching Lake Mead on the Arizona–Nevada border, where it turns south toward the international border.

Experts have stated that the Colorado River within the boundaries of Grand Canyon National Park drains an area of approximately 41,070 square miles. The major perennial streams feeding into the Colorado are related to large perennial spring systems on both the north and south sides of the Canyon. But, the majority of water sources are intermittent or ephemeral in nature. The availability of water in these individual systems is closely related to geologic structure, seasonality and annual precipitation. Knowledge of every water source within the Grand Canyon is still incomplete.

More Info: en.wikipedia.org