Can I replicate this "bumpy" fabric on a pillow?
You can't sew fabric like that from normal flat fabric.  If you have the fabric, you can sew it together as a pillow.  But your question seems to be about making the fabric itself.  To make fabric like that, you would need to weave it on a loom.  (If you have never woven, my answer will make no sense at all.)
Here's how to do it.
Cast on a very sparse warp; about 3 threads per inch by the looks of it.  These are the thin threads running horizontal in your picture. The weave you would do is called a log cabin weave.  For your weft, get two different kinds of bulky yarn.  You will alternate in a pattern like this:

A pedal: go left with the first color.
B pedal: go left with the second color.
A pedal: go right with the first color.
B pedal: go right with the second color.

Repeat this pattern throughout the whole fabric. Make sure not to beat the weft down too hard.  This is a very loose weave.