Nope. You can work out how many transfers you have using this: post354376.html#p354376 (linked in siggy also!) But unless you keep track of whats used, then no way of knowing whats left.
Darn lol I guess I should pay more attention. I just have so many accounts sometimes I forget. I should make a notepad document for it or something.
Well if you know how many transfers you get a month, there is a way of keeping track of what you've used as far as outgoing transfers go. There's a link for it but I don't know it right now. It's the transfer status page, and it tells you "(insert username here) has adopted (insert pet name here)". If nobody else comes along and posts it before I get home tonight I'll find it and edit this post.
It will also tell you on the confirmation page before you officially transfer a pet how many you have a month (i.e. This is one of your X transfers a month) but it doesn't tell you if you've used any of them.