What does the sewer QR code return

There is one room in particular that provides what you're looking for. You don't need to copy NPC dialog, though part of the fun is copying the other wall-writings for later consideration. So there is no single intended method (I imagine most people will look online)—your method could work, though it sounds frustrating.

Want a hint for that room's location?

It's off of the waterfall room.

Want an even bigger hint?

A pillar in the room links the code to an English pangram (sentence containing every letter of the alphabet).

And to give it away:

In the tree room above the waterfall, a fox jumps over a dog next to this pillar, which reads all the way around: "The quick brown fox jumps over the lazy dog." (image)

The 16-cube area is the ancient version of the temple/science warp zone. That QR code was under construction long ago, but it now appears "fully realized" on the same wall in the temple/science warp zone. So it can be completed—indeed, it has been.

If you're curious, the QR code links to...

text: "RT RT LT RT LT LT LT RT" (which triggers an anticube, as shown above)

This is a higher resolution of the screenshot, from the PC gersion of the game:

I have then extracted the QR code, converted it to black and white, removed Gomez and some dust, and applied a median filter. This is the complete QR code, which you can scan on a website.