There are at least four layers to this problem.
First, the QR code creator has to be able to handle the characters involved, and many are limited to the simple ASCII set defined in the older QR code standards.
Second, the code protocol has to be able to handle the characters. If you're using the built-in vCard form, it's limited to v2.1, which very much has the problem above and a few others. You might have more success using either vCard 3, or the BizCard protocol, which (I understand) is much more common in Japan and Asia. They are similar but it will all come down to the reader/parser—
The reader (on the smartphone, etc.) has to be able to read the encoding and extract the letters, and sometimes needs to be able to parse a specific set of protocols (instead of just pulling an unprocessed string of data from the code).
And finally, whatever app is receiving the data — a raw QR code reader, a Contacts app, something more sophisticated — has to be able to parse the protocol and the characters.
With Kanji, the problem could be in one or more of the above steps. I'd suggest you not use InDesign's built-in forms, but construct your own data protocols in the Plain Text mode. From there, I'd be sure to follow either the vCard 3 or the current BizCard format. For proofing and testing, use a generic QR code reader that will simply present the data decoded, and not automatically try to load it in Contacts or whatever. And then, make sure whatever destination app you wish to receive the data can handle the extended character set.
Here's a general primer on QR codes from InDesign: http://nitrosyncretic.com/DPR/dpr_qrcodes.php. It has links to both the QR code standard (which might have some answers about character encoding) and the vCard standard and protocol structure. You'll have to find info on BizCard through a search.
Hope that helps.
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