Sure

I managed to transfer two of my pets from my old account to my new one, and another pet to my friend.
There's two methods, but both have the same idea. What you need to do, is either have two browsers open (eg, IE and Firefox), with one logged into each account. If you're tranfserring to someone else, just make sure they're logged in.
On the account you're transferring the pet to, put this URL in the address bar:
http://neopets.com/process_adopt.phtml? ... ETNAMEHERE -- change the PETNAMEHERE to the name of the pet you're transferring. It's case sensitive.
Get the person to load the page, and keep refreshing. (If you're transferring by yourself, just hit enter so you have the error.) Then hit abandon, and as soon as the page starts to load, refresh the page in the account you're transferring to. Hopefully it works!