Can a Thoroughbred crossed with a Quarter Horse enter a race?

No, Thoroughbreds come from Turks, Barbs and various Arabian sires and dams. The British had founded the breed as a hybrid of other fast breeds. The Thoroughbred racing industry in North America only allows 100% Thoroughbred bloodlines, parentage assured by DNA testing and registration. Quarter Horse racing is a separate industry. == ==

No it can't have any QH blood to run on the JC track. An Appendix QH can run with AQHA if registered though.