Flexible room treatment material???

IDK about a quilt but you can get acoustic tiles made from fabric/polyester fiber that you can attach to the top of the hearth. There are also tiles that look like art work.

https://www.amazon.com/Soundproof-48X24Acoustic-Absorbing-Recording-Dampening/dp/B09HGZQFX5/ref=sr_1_32?crid=1NZQALGOJGFQ&keywords=troy+studio+acoustic+panels&qid=1679507887&sprefix=troy+studios%2Caps%2C159&sr=8-32&ufe=app_do%3Aamzn1.fos.f5122f16-c3e8-4386-bf32-63e904010ad0

Hanging actual quilts would probably look a lot better.

Consulting with an interior decorator should be helpful.

They do make diffusion you can hang, a good wall tapestry is always an option. :)

A tapestry or rug would look nice. Staple a board to the back along the top and use L brackets to hang from the hearth ledge?

OK. Thanks.to all. 

The consensus seems to be "an ordinary quilt/tapestry/rug".  

@kota1 

I was thinking of the front face of the hearth, rather than the top. 

@stuartk 

I had to hang a diffusor near my window, I see what you mean. You can see my virtual system for a pick, it worked out well and didn't block the light either.