Not unless your computer is already compromised, and in that case it would be easier for the malware to just phone home. That's the long answer, the short answer is really: no.
It's just a text file and doesn't hold much information (each request to the server that originated the cookie sends the cookie, so by necessity it needs to be very small). Usually it will just contain a unique numeric id that the site uses to reference data stored at the site.
For example, a2k uses a cookie that just contains your session id, which is just a long random string of numbers and letters that is tied to a session on our end.
The way the internet is designed is "stateless", which means the core protocols weren't meant to store "state". Each request was anonymous and blind to previous requests.
Storing "state" means stuff like knowing that you are logged in (instead of requesting a login on every page load) and cookies are just a way to store application state.
So as an example, when you log into a2k it sets a cookie with your session id, and with each request the server looks for that session id and if it exists and matches what you sent it keeps you logged in. There is another cookie (autoLogin cookie) that stores your login to automatically log you in if your session dies.