Plotting directly a dot file requires to be able to display the nodes in the correct positions, and link them properly. This is done by very complicated engines and is not included in the dot file, which is simply a description of the graph. Maybe there are C++/Qt libraries that take dot files as an input, but I don't know them.
According to this topic (Graphviz: How to go from .dot to a graph?), you can easily transform a dot file into a png or svg file. If you want to display the rendered graph into your application, the easiest thing to do is to generate that png picture (from the shell or from your code) and to manipulate it from your program.