Hello BuraCULa,
You've run into a limitation of the plug-in. The following is from the FBX documentation:
"There is a known FBX limitation where exported binormal and/or tangent information does not appear, even if you activate the Export Include Geometry Tangents and Binormals option.
The Tangents and Binormals option only works on meshes that have only triangle polygons, so you may need to triangulate the mesh before activating this option. "
The geometry has to be triangulated before export (all diagonal edges must be displayed on the mesh before export). If it isn't already, you can make it an editable mesh, or as you saw, leave it as default, as long as it is in triangles. After importing into Max or Maya, you can try and run the Quadify Modifier in Max or use Quadrangulate in Maya to return you geometry to quads.
Regards,
Carlos