A home page is something you define server side. Media queries run client side.
However, I can think of three ways of faking a similar behaviour.
Scenario 1: you are thinking of making many changes based on mobile devices. In this case it makes more sense to create a new mobile theme. You can perform server-side redirections on the mobile theme or do whatever you want. You can set it in admin/general.
Scenario 2: you are thinking of making a few amount of changes based on mobile devices. In this case it would make more sense to reuse your current theme and adapt it in a similar way as before. However, you won't know if the requesting device is actually mobile but you could at lest call the qa_is_mobile_probably() and decide the content to display on the fly. Note all these changes are server-side as well.