I did a deep search and discovered exporting a geometry node animation was not possible in the last version, was this added in 3.1? I haven't been able to get it to work if it has been added.
When searching for features that have been added, you can start with the References/Release notes for the new version, in this case 3.1. Here are the sections where such an update might be mentioned in the case of Geometry Nodes and Animation:
Working backwards, the only update to FBX export is a speed improvement. While considerable work was done on Geometry nodes, none of it appears relevant to FBX export or animation, except the addition of the Scene Time node but that would have no impact on export. None of the animation changes are relevant.
So no, no change has been added that would allow such export.