Please post a photo in this thread so we can get a closer look. I do not think it looks like a moiré. I think it looks like a jacquard weave - probably a damask. The pattern is called a "flame stitch" but the actual fiber could be cotton, silk, rayon or a synthetic or a blend. It's harder to tell when something is made into a headboard or pillow because you cannot get a loose thread to do a burn test, and you can't examine the whole thing. Do you have any of the fabric that is not made up? If so, please show the reverse side, also.
Thanks.