1. SSH into your old server. You should be able to find these two directories:
/usr/share/ssl/private
/usr/share/ssl/certs
The first directory contains the key file while the second contains the certificate signing request (CSR) and the certificate itself.
2. Now identify the relevant files you'll need to copy over to the new server.
If the SSL domain is "secure.domain.com" these files should be copied over to the exact same directories on your new servers:
/usr/share/ssl/private/secure.domain.com.key
/usr/share/ssl/certs/secure.domain.com.cabundle
/usr/share/ssl/certs/secure.domain.com.crt
/usr/share/ssl/certs/secure.domain.com.csr
3. Now use FTP,WGET or your preferred file transfer method to move these files over to the new server
4. Log into WHM as root, go to SSL/TLS -> Install an SSL Certificate and Set Up a New Domain
5. In the Domain field, enter the relevant domain (eg. secure.domain.com), click on the fetch button ABOVE (NOT below). Your SSL cert should be displayed in the box and the username, IP fields should appear automatically.
6. In the panel immediately below click on the FETCH button to fetch the .key file. This should appear as well.
PS: You might need to double check the crt. Compare the crt on both server. It might need a copy and paste the crt.
