Much to my dismay, I failed to find any PCBs on the internet that would serve to adapt the Q20 keyboard to pins suitable for prototyping. However, the Q20 is a component of the Solder Party Q20 handheld keyboard, which has all the hardware schematics in KiCAD format here. I was able to extract the symbol and footprint info from this schematic, save it, and build the following below. As this was my first experience with KiCAD, it took about six hours (lol) to figure out how precisely to get this to work. The secret was redoing the symbol with all pins, even the hidden ones, connected to a 1x20 pin connector on either side.

Once all that was done, it was simply a matter of organizing the footprints and cutting the board into some sort of compatible shape. I separated the two columns of pins by 17.5 mm, which should provide 1 or 2 breadboard connections per pin (the pins should go into the third hole from the center of each row). I ordered the board through OSH Park (www.oshpark.com), 3 for 10. They look pretty good too! I added the KiCAD files for anyone who wants to order them.