Closeable and AutoCloseable are general-purpose interfaces that do not know anything about flushing. So you can’t find any information about it in their documentation – except some words about releasing resources.
A Writer on the other hand is a more specific-purpose abstract class that now knows something about flushing. Some excerpt of the documentation for the method Writer.close():
Closes the stream, flushing it first.
So – yes – when using a writer, a close will always also flush. This basically means that you have to consult the documentation of the concrete classes that you are using when trying to find out what closing really does.