If you want to go closed source and distribute your application, you can't use GPL. You can use LGPL when linking either dynamically or statically. For the latter, you need to provide object files for all of your code, so that they can be statically re-linked with Qt.
Conercerning your base issue, Qt 5 includes a serial port module. On Qt 4, you'll need a platform-specific solution. Reducing the scope might help to point you to something. I suggest that if you don't target portability, you should add relevant tags and comments. Otherwise, saying that you need a portable solution is a good thing too.
As far as I know, on most paltforms, accessing to serial port requires higher privileges, so be ready for that in your deployment.
Concerning qtextserialport, license is BSD so it would work for your clsoed source application. If you post a compilation issue on where it fails, maybe people will be able to help you
I tried to compile it using VC2010 and Qt 4.7.2 and head of project (I just cloned it) compiles flawlessly: I got a nice qextserialport1.dll (that I admit I didn't try using).