Is copyright applicable to popular quotes and how to post them on your fb page without copyright iss

Copyright law differs depending on where you go although a number of countries have reciprocal treaties. In the US, the question is one of fair use. This is a four-pronged test:

1. Amount and substantiality of the original material being used (less means less likely to be an infringement)
   
2. Purpose and character (educational or newsworthy are least likely to be seen as infringing). This is also referred to as whether a work is transformative.
   
3. Nature of the copyrighted work (it's less likely for factual versus fictional works, for there to be an infringement) and
   
4. Effect on the market (while disseminating information for free or not isn't really a test, it's more likely to be considered an infringement if you're making a buck off it).

See: https://fairuse.stanford.edu/overview/fair-use/four-factors/

So you're probably okay with reference to the people who said the quotes. But as for the folks who lifted the quotes and slapped them on their websites? If you're using everything but the background then you could be failing the substantiality test -- but keep in mind those sites might not have a copyright leg to stand on to begin with. If you're taking advertising dollars from them, then that won't be so great, either.

An old famous quote from, say, Mark Twain, is in the public domain anyway. But I'd be more cautious and not just copy/paste.

Note: I am a lawyer (retired), but I'm not your lawyer. Plus I'm only talking about the United States. Laws will undoubtedly differ elsewhere. If you are concerned, please seek a qualified legal practitioner in your area and, yes, pay them for their advice.