From reading the GOG store listing, The Elder Scrolls III: Morrowind GOTY Edition does not appear to support cloud saves. This appears to the only feature it is lacking when compared with the Steam version. Cloud saves should be added so GOG's version can be competitive with Steam's version.

You can always contact support. I think people have done it in the past and they added cloud saves for other games.

To me, the GoG version is far superior, even without "cloud" saves.

With the Steam version, you still have to ask Valve for permission to play "your" fully offline, singleplayer game.

But is it really yours if you have to ask permission to play it?

Not that it matters much: The Morrowind game itself doesn't support cloud saving, not in GOG and not in Steam.
The cloud saving function for Morrowind in Steam is a Steam add-on: Steam manages the native Morrowind savegames, transfers them between disk and cloud in both directions, synchronises disk and cloud saves, etc.
As GOG offers the original Morrowind game, you can't expect cloud saving for the GOG game. If at all, cloud saving for GOG Morrowind could be offered via Galaxy (which I can't comment on because I don't use it).

What I wonder though, why is Morrowind the only main line TES game here, that does not support cloud saves?
Even TES I&II that need DOS have Cloud Saves. Only Morrowind, Battlespire and Redguard has no Cloudsave feature.
While I can understand why the work for the side games is not worth it, it is baffling that Morrowind is not considered worth the work.