Well, we started about 2 years ago and our company's been calling it Q internally. This is just the front end SDK for what we built (Q spans PHP, Node.js and front end css/js.) I think we're going to keep the name.
This is the biggest piece of feedback you have gotten yet. Q isn’t nearly as famous as e.g. Backbone or Ember or Modernizr, but it is established and well-known. Your name is actually a barrier to this catching on, I think.
This is where you ask yourself "would I like if someone releases a product with the exact same name as my company's and moved us to the second page of Google?".
Even if that was somehow relevant (it doesn't really matter what you called it internally if you didn't use the name publicly), the existing Q.js is older!
that is a bit childish. the time you began development is not relevant, and seeing as q.js is already popular, it is safe to say it will stay that way.
changing the name is great advice. consider it.
As a long time user of the original Q (which has been around since at least 2009 [1]), I'd also suggest picking a different name for another reason: searchability.
OK sorry let me put it another way ... the larger library is meant to run on PHP and Node.js, and the Q.js and front end part is just the SDK to use it on a web browser. So when the whole thing is released, it will mostly be a PHP+Node.JS+Web library.