No, the Colorado River does not flow through Texas.
The Colorado River (Western U.S.) rises in the Rocky Mountains of Colorado, flows southwest through Utah, Arizona, Nevada, and California, and empties into the Gulf of California (Mexico). Its basin covers parts of 7 U.S. states (CO, WY, UT, NM, NV, AZ, CA) and 2 Mexican states—but not Texas.
Common Confusion: Texas has its own Colorado River (of Texas)—a completely separate river that rises in West Texas (Dawson County), flows southeast across central Texas (through Austin, Bastrop, Columbus), and empties into Matagorda Bay on the Gulf of Mexico. It is ~862 miles (1,387 km) long and wholly within Texas.
Two different rivers. Same name. Zero overlap.