Legit one of the most annoying things I've ever come across with this service.
You using Galaxy?
Nice oversight in interface design GOG. *headdesk*
From what I've heard from others (so, take this with a grain of salt, since I'm not a Galaxy user myself), there's no way to access serial keys from your library within the client; you have access GOG via a normal browser, and look the game up in your account that way. They're always(?) under the "More" dropdown, under "serial keys" (naturally).
Yeah, thanks a bunch. Found em, that is rather unintuitive on GOG's part...