I have a Bedford stone chimney and I really dislike the stone. Can I brick over this stone? If not, can anyone suggest what I can do?

If it's the color that you dislike, there are stains available in a wide range of colors that you could apply the the stone. I've used Chemstain by L M Schofield & it works great.

First question is, what is "Bedford stone"?

If the stone is fairly regular, without a lot of ins and outs, you probably could install thin brick over it.

To install real brick, you'll need something to support the brick...either a steel shelf angle expansion bolted to the chimney, or remove the stone and build the brick on whatever supports the stone.