There's not really that much of a Japanese population here.
Japanese don't tend to accumulate in ethno-centric little pockets. Most that come to the West do so as individual students and then professionals, research associates, etc. Toronto and Vancouver Canada both have fairly well-sized Japanese populations but as I said you won't find them clinging to one another in the same way as other groups. That's not meant to sound derogatory but let's call a spade a spade shall we?
If Seattle is anything like Vancouver (and I understand they are similar in many ways) there should definitely be some Japanese folks there since its a main westcoast portal from the East. While there may be no discernable 'Japan town', if you turn a few stones over you will find them at least socializing together on a semi frequent basis.
To start, internet based language exchange introduction sites, university student centers advertising English study, etc., will be likely places you may run into Japanese ESL students who wish to study. After that as you network outwards, you'll find that although they live independently and kind of scattered there are certain pulse points they're all linked to. From there you'll find the more socially oriented events, outings, groupings, etc. For example group outings to public areas for BBQ's (well summer's over but, just as an example), a group might accumulate and attend a festival or exhibition of some kind or to go to a well known landmark or something, kind of like field trips. You may find yourself doing a lot of really touristy stuff.
That was my experience in Toronto anyway, I found tons of Japanese folks to hang out with and study with.
Another tip, Chinese and Japanese don't tend to mix well it seems but Korean and Japanese young people really get along and co-mingle, again just my experience in Toronto. If there are Korean areas in Seattle you may very well run into some Japanese cats there.
I found these after a quick 1min google search:
Learn Japanese Seattle, Japanese lessons Seattle, Japanese classes Seattle, Study Japanese Seattle
http://www.seattlejapaneseschool.org/
http://www.seattlelanguageexchange.com/