I have SSH access to my university account using SSH without password. That's working so well that I even forgot that password altogether.
Now I have a new computer and would like to access my university account from there. But I can't without the password! Can I transfer the password-less authentication from my old computer to the new?
Of course, I could also ask the university sysadmins to create a new password for me, but they are known to be very slow, so it might take a few days before I can access.
This is not a site for end users, it is for System Administrators (et al) only.
That said, your "passwordless SSH" is actually an SSH Key Pair. The keys are just files on your computer; usually named id_rsa or id_dsa. If you are using PuTTY to connect, it's a file ending in ppk (PuTTY will have the path to the key file in it's configuration, look there).
Copy the file to the new computer and configure your software to use the key file. How exactly you configure your new computer will depend on the software you're using.