I have created an Ubuntu applications using Quickly, but I am wondering if the .deb package that Quickly has created will work on Debian. Do Quickly applications work only on Ubuntu?
This is largely application specific.
An application created in quickly will work on any Linux distribution so long as its dependencies are met. Quickly is just a method of creating applications.
If your application uses a version of a library not supported in Debian, then it would be necessary for you to provide this library yourself, either packaged with the application (and referenced accordingly), or in a PPA (overwriting the local version, or alongside the local version).
Also, any application that makes use of Unity/Ubuntu specific features will likely work, but only if those features are not made hard requirements. If for example you use an app-indicator for your application, it will work on Debian and other distros by using the fallback.
If you use Unity progress bars or other forms of integration, you would need to ensure that you make them optional features, otherwise your application will encounter errors.