Yes, mark an in point and out point on the time line and use the “export” command or perhaps even the “share” command, to render out a “self-contained” movie, in h.264 mov format… Then upload that movie to your facebook page.
I find that sometimes FCP gets quirky about knowing which part you want to render out, so personally, I make the end point first, then the beginning in-point, leaving the play head at the in-point , before I do the export.
This advice was for users of FCP7; if you’re on FCPX, go to that forum for specific advice.