Navisworks product can export whole model to FBX.
There is not a direct way to export some model items only. However, it can be workaround by hiding unnecessary items, and exporting. Then, only those visible items will be exported.
The snapshot below is after the FBX is translated by the services of Autodesk Forge, and rendered in Forge Viewer. I simply played with the Autodesk app that is based on the related technologies of Forge: https://a360.autodesk.com/viewer/
From API perspective, no direct method to export FBX, but you can find out the built-in plugin of Exporting and execute it. The code snippet below is a demo.
public override int Execute(params string[] parameters)
{
//********
// hide unnecessary model items. remain those items you wanted to export
//********
PluginRecord FBXPluginrecord =
Autodesk.Navisworks.Api.Application.Plugins.
...