This depends on the kinds of certificates you want to use. You tagged pgp, but I'm not sure whether you actually are looking for S/MIME (X.509) certificates.
Like Firefox, Thunderbird uses (unlike most other applications) its own certificate store and does not rely on the system's one.
- open Preferences (in the Edit or Tools menu, depending on your operating system)
- open the Advanced tab, then Certificates
- in the View Certificates menu, choose You certificates
- select the certificates you want to export and press Backup, or choose Backup All
On the new machine, open the same certificate manager, but choose Import instead.
If you use OpenPGP through Enigmail (a Thunderbird addon) and GnuPG, keys are managed by GnuPG.
gpg --export-secret-keys [key-id] > secret-key.pgp
to export your key on the old machine, transfer the file to the new one, and import it using
gpg --import < secret-key.pgp
on the command line. If you prefer using a graphical user interface of your choice, it also should provide similar functionality, but make sure to also export the secret keys, not only public ones.
Alternatively, you could simply copy the whole
~/.gnupg (unixoid systems) or
%APPDATA%\GnuPG (on Windows) folder to the new machine, this will also transfer all other's keys and trust settings.