Yes, you can share the terrain info if you export your tileset to an external file (using the small Export button below the Tilesets view). You can then add this external tileset to other maps using Map → Add External Tileset….
This functionality is getting a major revamp for the next version of Tiled to make external tilesets the default way of working with Tiled. Depending on your operating system I could point you to a build that you could already give a try. See also the latest development update.
You’re very welcome!