Older versions of MotionBuilder (versions 2010 and earlier) do not support hard/soft edge normals and can result in shading issues in MotionBuilder.
If you are using an earlier version of MotionBuilder and this problem occurs, activate the Split per-vertex normals option. When you activate this option, geometry vertex normals are split based on edge continuity. Vertex normals determine the visual smoothing between polygon faces. They reflect how Maya renders the polygons in smooth shaded mode.
Do not use this option if you do not require hard edges in MotionBuilder, since this operation duplicates vertex information and converts the geometry itself. You can use the Combine per-vertex Normals option in the FBX Importer to perform the inverse operation, but this may result in incorrect UV texture maps, see the note that follows this entry.
If you are not using a merge-back workflow (merging FBX into existing scene) using this option may result in incorrect UV mapping when reimporting the FBX file into Maya. Import your FBX file into to your original Maya scene, to avoid geometry UV issues with split geometry.
Exporting with the Split per-vertex Normals option disabled produces the following result in MotionBuilder:
Exporting with the Split per-vertex Normals option activated produces the following result in MotionBuilder: