You don't say what format the drawings are in, so that's where to start. After that, it's a question for QGIS people how to import data that is in whatever particular format the files are in.
If those drawings contain projected data and were saved into a format that does not automatically capture projection info, no GIS package can automatically assign the projection. You have to tell it what projection the data uses, because the format won't do that for you. The classic example is DXF format, a CAD format that doesn't save projection info.
If you save projected drawings into DXF format, to use that as an example, when you import those into *any* GIS, you have to tell the GIS package what projection it is supposed to use. That goes for Arc, Manifold, Q, and any other package.
That's a good reason to avoid using such formats or, at least, to make a point of jotting down somewhere the projection(s) used, so that on import into some GIS package in the future you can tell the package what projection it should use with that data.
If that is what's going on, carefully search your notes and try to remember what projection(s) you used. Read up on Q documentation carefully so you import the drawings correctly, and then use whatever procedure Q has for assigning projections to data from a format that does not automatically convey it.