This use is actually a little ambiguous. "the central country head-quarters" could mean either "the central headquarters of the country" or it could mean "the headquarters of the central country".
However, in English, it is more common when talking about a country to use the name of the country itself, rather than just saying "the country", so if it was talking about the "central headquarters" for the whole country I would expect it to say something more like "central American head-quarters" or "central U.S. head-quarters" instead, so my guess in this case is that it may actually mean that it's the headquarters for the "central country", that is the middle part of the U.S.
On the other hand, I note that the source was written some time ago (by Arthur Conan Doyle), so the conventions may have been different back then, I'm not sure. In which case, my guess may be entirely wrong.
So it really isn't entirely clear, and the only way to be sure is probably to find some more information somewhere about what sort of "head-quarters" Lilydale actually was to the American Spiritualists at that time.