Oh yeah, they do! Koi and goldfish were the first domesticated fish. There are some Chinese fish whose lineage has been recorded for almost 1000 years, like in the stud books for horses and dogs. Koi and goldfish are not the same species, though.
Originally Posted by Wiki
Goldfish and koi were developed from different species of carp; although they can interbreed, their offspring are sterile.[3] Goldfish were domesticated from the the Prussian carp (Carassius gibelio), whereas koi are the domesticated version of the common carp (Cyprinus carpio). In general, goldfish tend to be smaller than koi, and have a greater variety of body shapes, and fin and tail configurations. Koi varieties tend to have a common body shape, but have a greater variety of coloration and color patterns. They also have prominent barbels on the lip. Some goldfish varieties, such as the common goldfish and shubunkin have body shapes and coloration that are similar to koi, and can be difficult to tell apart from koi when immature.