The only bio to fbx conversion we know about is export of 3D content between Autodesk Mudbox 3D format to FBX format, something that should be actually doable with the Mudbox itself.
We do not know about any tool that can work with Mudbox's .bio files, and FBX is the usual export format for this type of content anyways, so using the original software is most likely your only option. You can always try to reach out to the Mudbox support or community though.
Updated: October 23, 2023