With the Steam Deck only a few days away from release, it seems people are wondering the support status of other stores. Seems GOG won't be supporting it, which isn't surprising.
Currently, GOG does support Linux with direct downloads, although they only officially support a few versions of Ubuntu. They do not, however, support GOG Galaxy on Linux. This has been a bit of a problem, as some games with multiplayer rely on Galaxy meaning a few Linux builds on GOG came without multiplayer.
Speaking on Twitter, in reply to a YouTuber asking if they will support the Steam Deck they replied:
There's no official GOG support for SteamDeck. But since the device has an open architecture, as we understand, you can install Windows and as such also run GOG games.
They never replied to our email enquiring about it.
Technically, you won't need to do that. GOG Galaxy can run in Wine, although unsupported. On top of that, projects like Minigalaxy exist, Lutris has GOG download support, Bottles has a GOG installer and even Heroic Games Launcher is working on Linux support that's close to release - so there's options available on SteamOS 3 with the Steam Deck to get GOG games. However, Galaxy features are generally locked to their own app.
One reason that's likely GOG aren't working to expand their Linux support to Galaxy and Steam Deck, is that they aren't exactly doing well financially.