A quine is more similar to (but not quite the same as) an uncompressed image containing a QR code with a compressed copy of the same image, which has its own challenges; but like, a cryptographic hash function (such as SHA2) is designed explicitly to make this hard... if you had chosen a simple checksum instead it would be trivial to create an image containing the text of a checksum of the image: a compression function (particularly a tiny custom one designed for this one time use, that removes barely any entropy at all) is predictable enough to be manipulated. (edit: I hadn't looked at this one, though; but now that I see its code, apparently it is just cheating by accessing its own code via the DOM; it doesn't need to do that, and it doesn't even really feel like a "quine" to me given this implementation.)

Clever use of Javascript.

  <a><script src="https://itty.bitty.site/lzma/lzma_worker-min.js"></script> 
 <script>a=document.querySelector('a');LZMA.compress(a.outerHTML,9,function(r){a.innerText=a.href='https://itty.bitty.site/#/?'+btoa(String.fromCharCode.apply(null,new Uint8Array(r)))})</script></a>

You can create videos on YouTube too that contain it's own URL.

As others have pointed out, it's not quite a quine and just some silly JavaScript that reads out its source from the DOM and feeds it back into a copy of Itty Bitty's URL generation algorithm.