Is her profile public? or can you only see it because you are facebook friends with her?
Are you a manager?
If she has in fact put in a formal complaint then management presumably knows about her allegations, so the issue comes down to whether or not what she has said online is libellous/likely to damage the companies reputation and if so, whether it is publically available (e.g. would someone reading the public forum, but not being facebook friends with her, or already aware of her situation, be able to identify the company?) If yes to both, then I might consider taking a screenshot of the forum posts, printing it off and sending it to HR - if it is available publically you can do this while you're not logged in to facebook so you won't be identifiable. However, from what you've posted, it doesn't sound as though the majority of people on the forum will have any idea who she is or what the company is, and you don't (from what you say) have any knowledge as to whether her claim is valid or not.
But i don;t think you have any obligation to say or do anything at all, unless your employment terms require you to report incidents involving the company's reputation.
I would see it differently if she was posting stuff which would be directly relevant to any claim she made - for instance if she was posting things which made it clear her claim was false.