Drawable and Context exist so that LWJGL can provide different implementations of GL contexts, they are not meant to be used outside the library.
What you probably want to use is GLContext.useContext. This method takes a plain Object as its single argument, which is supposed to be a GL context that has been created outside LWJGL. After a call to useContext you can use the LWJGL API to submit GL rendering calls, with the limitation that the external context has to have been created and made current in the same thread that called GLContext.useContext (that's a GL/driver limitation, not LWJGL limitation). Something like:
QGLWidget widget = ...create widget...
widget.makeCurrent(); // Imaginary call, you need to be sure the widget's GL context is made current before useContext.
GLContext.useContext(widget);
...
GL11.glFoo(...);
GL11.glBar(...);
...
widget.update(); // I assume there must be a way to swapBuffers or something.
This is all assuming QGLWidget creates and manages its own GL context (sounds like it from your description). If you need to hack your own context in there, then I guess you'll have to checkout LWJGL, implement your own Drawable and use that custom build in your project.