Can I use a quote on the internet for T-shirt design?

I depends where the quote came from. Was it an excerpt from a written publication? You do not have to steal an entire writing to infringe upon ones rights. Case law shows that often the use of brief quotes 75 words or less can constitute infringement. Damages are another thing. is the quote a slogan or is it part of a writing? How old is the quote? There is a lot of facts that i would need to know to render an opinion here.

Just because you see something on the internet without the name of the author, please don't assume the quote isnt copyrighted and the owner doesnt care. Much of this sort of thing has been stolen and posted and reposted without permission. Try to track down the original author and when the quote was created. If it was prior to the early 1920s, its probably in the public domain and you can use it without permission. Anything created after that is still copyrighted; you should obtain the copyright owner's written permission and cite the authors name with the quote on your shirts.

You should consider having an attorney make sure there are no copyrights in effect for all or part of the quote. Additionally, it would be wise to determine whether all or part of the quote has been registered as a trademark in the U.S. If you are an Ohio resident, my firm would be able to assist you.

Chances are you are fine using it if there is no known author. If the author pops up at some time and is able to prove authorship, you may have an issue.

Maybe. The question is who - if anyone - might claim ownership of the quote. OLD quotes - e.g., those over 100 years old - are likely in the public domain - free for anyone to use.