The hashcode of a given String
object does not depend on the locale. That should be obvious from the javadoc that you linked.
However, any transformation that produces different characters in the string will lead to a different (non-equal) string and a different hashcode. For instance, translating a bunch of bytes to a String using different default character encoding can result in different characters.
Summary, changing Locale doesn't directly affect String hashcodes, but it could cause your application to produce different String values, and THAT will affect their hashcodes.