Alas, both words most probably do not come from the same root. So far as we know, their identical spelling is completely accidental. They were already spelled the same more than two thousand years ago by the Romans.
Our word German comes from Latin Germanus, first attested in Caesar, which was used to describe the Germanic tribes by the Romans. According to the Oxford English Dictionary (2nd ed.), it may be of Celtic origin, as a name used by the Celts to designate their neighbours; but there appears to be no consensus yet.
The word germane comes from Latin germanus (adjective "related, relative", noun "brother") This comes from the Proto-Indo-European root *gen-, which means something like "give birth". Our word germ, from Latin germen, "sprout", comes from the same root, as does genetic, through Greek genesis, "birth".