Did you make a "thick" terrain (with some kind of base under the mesh), or just a modified plane ? If it's the latter, it's no surprise that it is imported as a plane. Try adding 4 vertices in blender, under each corner of the map, select all vertices, press F (to create the missing faces) and then use the "triangulate faces" option (right-click menu with face selection enabled)
As for the material, as long as I can tell from experience, Godot can not import materials along with .obj objects, so you have to re-make every material in Godot and re-bind them to each material slot of the .obj.
Exporting in gltf seems to solve this issue: the material remains attached to the mesh.