<p>We currently maintain a suit of MFC applications that are fairly well designed, however the user interface is beginning to look tired and a lot of the code is in need quite a bit of refactoring to tidy up some duplication and/or performance problems.  We make use of quite a few custom controls that handle all their own drawing (all written using MFC).</p>

<p>Recently I've been doing more research into Qt and the benefits it provides (cross-platform and supports what you might call a more "professional" looking framework for UI development).</p>

<p>My question is - <strong>what would be the best approach to perhaps moving to the Qt framework</strong>?  Does Qt play nice with MFC?  Would it be better to start porting some of our custom controls to Qt and gradually integrate more and more into our existing MFC apps? (is this possible?).</p>

<p>Any advice or previous experience is appreciated.</p>

(This doesn't really answer your specific questions but...)
I haven't personally used Qt, but it's not free for commercial Windows development.
Have you looked at
wxWindows
which is free? Nice article
here
. Just as an aside, if you wanted a single code base for all platforms, then you may have to migrate away from MFC - I am pretty sure (someone will correct if wrong) that MFC only targets Windows.
One other option would be to look at the
Feature Pack update
to MFC in SP1 of VS2008 - it includes access to new controls, including the Office style ribbon controls.