A very interesting question. I don't know that much about the various regional forms of traditional East Asian cuisine, but I'm sure that if the Japanese and Koreans colonised west coast North America, they would focus a lot on fishing at first. It would only be natural, given their domestic food-acquiring focus and the existing know-how they had with that. A greater shift towards trying out land-based new food sources would occur concurrently with efforts to expand more inland, eastward of the already chartered and explored coastal areas. I think the Chinese, being a nation with less of a focus on maritime cuisine, would be more willing to research and experiment with new ingredients from North America.
The northern as well as southern native cultures of the west coast would no doubt have an influence on what cooking and eating habits the longer-term colonists might develop over time. Provided their relationship with the natives is stable enough and hey can get along without too much overt conflict. Some cultural exchange is bound to happen, advice on food included. One thing to factor in is that there might be misunderstandings, such as some group of explorers or traders bringing back crops or berries or something from a native culture, but having confused (or misinterpreted) what they're supposed to do with these.