No.
It has nothing to do with serial keys, though. Even if GOG is DRM free, they often supply a serial key with their games. This is for your convenience. Should GOG one day go bust, you still have your serial key – which is your unique code, or "proof of purchase", if you will.
This should not be confused with DRM, though. DRM simply put limitations on how the game can be copied or transferred. A game with no DRM can still ask you to enter a serial key for a number of reasons.
A serial key should not be confused with a Steam activation key either. It's a separate concept. If you buy the game from a retailer, or from the developer directly, they will often let you activate it on Steam, so that it is connected to your account.
Some times, they may also give you a gift code for that specific game on GOG or other game platforms.
What you will not see, however, is Steam providing GOG codes or vice versa. The reason for this, is because they are competitors. It would be similar to buying a gift certificate at the Adidas store, which you can use at the Nike store – or Google Play versus the Apple Appstore. They would be a) cutting away the profit from their competitor and b) leading customers to the competition.
HOWEVER!
In some cases, Steam accepts the regular serial key as the activation key, but that does not include Don't Starve. You can see the list of games which accept third party keys here: https://support.steampowered.com/kb_article.php?ref=7480-WUSF-3601
Best Answer
Some games allow you to do this, most don't. The game "Prison Architect" for example does support this:
Theres however a second way to obtain your key, by looking for it in your Windows Registry.
Open regedit.exe (enter regedit in the "Run" window after pressing
+ R) and browse to:
In there you will find folders called
TestApp228760(TrackMania 2 Canyon) and/orTestApp232910(TrackMania 2 Storm). Inside those folders there will be an object namedSteamKeycontaining your key.