Without going through a Computer?
No.
Chances are pretty good that the books you currently own are in ePub format - correct? The Kindle cannot read ePub files.
Where did you obtain the books you currently have for your Sony? If they are DRM locked, you will have to remove the DRM. Then you will need to convert the file from ePub to a format the Kindle can read, most likely .mobi or .azw
Calibre is an excellent tool for handling that.
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