It works by checking a json file or gemfile. So I sent Github a message saying that it should be easily possible for them to also check the REQUIRE file for Julia packages for dependencies. They responded by saying they have forwarded this request to their team, but cannot guarantee anything.
So if you are also interested in seeing this feature, perhaps consider sending a support ticket to express your interest in having a Julia dependency feature on the Github website. If more than one person expresses an interest, it might have a chance of becoming a reality.