There are plenty of noodles in Korea, but your udon and ramen are clearly Japanese and your jjamppong and jajangmyeon are (ostensibly) Chinese. For a taste of true, home-grown Korean noodles, look no further than cold noodles, a.k.a. naengmyeon (냉면). The noodles themselves are typically buckwheat based, like soba (which is also often served cold in Japan), and come either in soup (물냉면 “mul naengmyeon”, water cold noodles) or dry (비빔냉면”bibim naengmyeon” mixed cold noodles). I believe the soupy version is more traditional.
There is a huge difference between the cold noodles from North Korea, called Pyongyang Cold Noodles, to distinguish them from the South Korean version. The South Korean style is loaded with vinegar (or kimchi juice), chili sauce, slices of pickled radishes, apples, and half of a hardboiled egg to create a cacophony of flavors that can take a little getting used to. The North Korean style is simpler, where one is supposed to appreciate the purity of the broth. The difference is a little too on the nose as a metaphor for the two Koreans.