The Berne Convention on copyright specifies the Right to Quote as an exception to copyright.
Article 10 (1) It shall be permissible to make quotations from a work which has already been lawfully made available to the public, provided that their making is compatible with fair practice, and their extent does not exceed that justified by the purpose, including quotations from newspaper articles and periodicals in the form of press summaries.
(2) It shall be a matter for legislation in the countries of the Union, and for special agreements existing or to be concluded between them, to permit the utilization, to the extent justified by the purpose, of literary or artistic works by way of illustration in publications, broadcasts or sound or visual recordings for teaching, provided such utilization is compatible with fair practice.
But as the second paragraph says, it is up to each country to determine in law how this right to quote shall be specified. Many countries cover this right under their broader rights of fair use or fair dealing. There are also several countries which have not signed the Berne Convention.
Wikipedia goes on to say that
National legislations usually embody the Berne Convention limits in one or more of the following requirements:
- the cited paragraphs are within a reasonable limit (varying from country to country),
- clearly marked as quotations and fully referenced,
- the resulting new work is not just a collection of quotations, but constitutes a fully original work in itself.
In some countries the intended use of the work (educational, scientific, parodist, etc.) may also be a factor determining the scope of this right.
This right to quote is most unambiguously applied to quotes in academic texts, where you can quote from something and then discuss, engage, refute it, etc. Most academics probably don't even know that the right to quote is specifically granted by the Berne Convention, it's just a simple fact of how academia functions.
What you propose to do in the question is not similar, and it sounds like you might be making what the third point there calls "a collection of quotations". If your web page consists of nothing but quotes from Facebook, without commentary or discussion, while it's not impossible that a court could rule in your favour, I wouldn't bet on it. If that's what you intend to do, make sure you consult a lawyer who can tell you whether your project would be within what your country's laws permit.