Generally yes, provided you do not imply sponsorship or endorsement by the author of the quote or any third party, particularly some business.
Many quotes are registered trademarks. You cannot use someone else's trademarks in such a way as to confuse a significant number of consumers as to source or sponsorship or control of your products. For example, you cannot put JUST DO IT. NIKE on T-shirts for sale without permission - people will think Nike licensed you and you will be subject to suit by Nike if Nike chooses to assert its famous brand. Even if you do not attribute the quote to it's source, there may still be an implied association. If you put IT'S THE REAL THING! on a shirt and sell it you will surely hear from trademark lawyers for Coca Cola. If you put ONLY YOUR HAIRDRESSER KNOWS FOR SURE you will likely hear from lawyers for Clairol.
Many quotes are associated with famous people. If you use the quote and put the name of the person, you are likely to be violating their publicity rights. For example, if you put YOU AIN'T NOTHING BUT A HOUNDOG! - ELVIS PRESLEY, you can be pretty sure of hearing from lawyers for the estate of "The King of Rock n Roll". Even if you do not provide attribution, although you are likely safer legally, you may still hear from the author's lawyer. You likely cannot afford that fight and will need to change and perhaps even pay a settlement. These celebrities have a royalty stream to protect.
Other quotes may be public domain due to passage of time. For example, you might put TO BE OR NOT TO BE, THAT IS THE QUESTION on a shirt for a gay rights promotion without much fear of trademark or publicity right or copyright problems unless some organization has already adopted that and registered it. A TESS trademark search at www.uspto.gov will quickly let you know if there is a major trademark problem with the usage, although there might, on rare occasion, be common law trademark rights not found in such a search.
The best and smartest approach in these situations is to consult an intellectual property law attorney for advice. Many offer free initial consultations.
Intellectual Property Law Attorney