An academic research paper enhanced my startup's potential. What do I owe the authors?

An academic paper published by a major university contains some methodologies that fit perfectly with my start up idea. I would like to incorporate the methodologies into my start up but I am concerned about the following things:

1. Do I need to quote the authors somewhere in my start up?
2. Do I owe them anything if the start up becomes profitable?
3. How do I know if I'm free to use published academic research to support my profits?

To answer you questions:

1. I think this depends heavily on whether or not you are publicizing your methods or whether its important to your users/regulators/etc that your method be known. If these methodologies are a critical part of your business and you want/need to promote that, than yes, you absolutely should cite the authors.
2. By virtue of publishing the paper, the authors are effectively inviting other people to try their methodologies. Unless the method has been patented, you don't owe them anything except your gratitude. And even if it has been patented, you probably owe the University money and not the authors.
3. The only way that these methodologies can be restricted once they've been published is if a patent has been filed on them. However, if I remember correctly (IANAL) I think they have up to one year after first disclosure (and this publication would count) in which to file the patent, so you should be a little careful here. That being said, even if they have patented it, the university will almost certainly be the patent assignee and you should talk to their Technology Transfer office about licensing the patent for your work. As a result of the Bayh-Dole Act, universities have been under a great deal of pressure to patent and license the work of their researchers, so assuming you are willing to accept having to pay a license fee to the university, I don't see this being and obstacle.

With all that in mind, it is probably worth consulting an IP attorney if you are really planning to incorporate these methodologies into your startup.

The only way to protect a methodology that is presented in a public paper is to patent it. If the university has a patent, then it could prevent you from using the methodology. Otherwise, you don't have any obligations.