Hi,
Within your Teamspeak 3 server (old computer) on the top toolbar click on Settings > Identities. A window will appear
As you notice you have several options. Exporting this ID will export your ID on all the servers you have access to. Exporting will allow you to save all these settings. Choose the identity you want to export (usually Default) then click Export on the upper right, then choose a filename.
On your new computer you will want to do the exact same thing but instead of exporting you will want to import. Go to Settings > Identities, and choose Import then point to the INI file you had just exported to complete the import process.