No, they don't. But there is one case in CSS1 where an alternative notation with quotes is allowed, viz., in the @import statement (see below).
Here are some examples of correct URL's:
BODY {background: url(pics/wave.png) blue}
BLOCKQUOTE {background: url(http://xyz.com/pirate) center}
@import url(sub/liststyles.css);
Here are some common mistakes:
* BODY {background: url("pics/wave.png") blue}
* BLOCKQUOTE {background: "http://xyz.com/pirate" center}
* @import url('sub/liststyles.css');
The @import statement allows an alternative notation, which does use quotes, but omits the url() part, for example:
@import "sub/liststyles.css";
@import '../common.css";