You sure can use MPI with Qt because MPI and Qt are two independent C++ libraries. But it may not make a lot of sense:
1. parallel computing is not GUI applications related. If you need to parallel any part of your code,
you need to separate it from your code and make this part stand-alone.
2. you may need to buy one Qt license for one node. Nobody likes to do this.
However, you can use OpenMP to speed-up your code up if you have multi-core computers.