I believe the information provided by ID software with their code releases would be the closest thing to actually asking ID yourself
ID Software github - readme.txt for Quake
All of the Quake data files remain copyrighted and licensed under the
original terms, so you cannot redistribute data from the original game, but if
you do a true total conversion, you can create a standalone game based on
--- snip ---
You can still download one of
the original quake demos and use that data with the code, but there are
restrictions on the redistribution of the demo data.
If you never actually bought a complete version of Quake, you might want
to rummage around in a local software bargain bin for one of the originals,
or perhaps find a copy of the "Quake: the offering" boxed set with both
The only company that can truly answer your question would be ID software, if they consider your game to be in violation of their copyrights then they will most likely take legal action regardless of what anyone here tells you. On the other hand, they have released the source code of Quake in a manner that allows you to modify it as long as you don't distribute the other files with your modifications.
You may wish to consider that by making the game "borrow" models from another game installed on the users computer you will never be able to make a full installation that does not require that game. In short, you will artificially shrink your potential user base, all to save yourself the relatively short time it would take to make a suitable weapon (and projectile) model in a 3D program.
Personally I would use Blender and make a quick weapon myself, even if it looks crude it would be safe to use and could be upgraded in the future without any legal dilemmas.