You can. I’m not particularly familar with Thunderbird, but you can export your key to system/user keyring then you can sign any data you want using GPG. However, I doubt tool exists for you to embed the signature to a PDF like x.509 signing would.

https://superuser.com/questions/653231/embed-a-gpg-signature-in-a-pdf-file#1361205

Do you mean using PDF’s built-in signature mechanisms? I don’t think so.

If you want to do regular PGP file signing you can export the key from Thunderbird and do whatever you want with it.

https://portablesigner.sourceforge.net/index.html

You may need to use your key to generate a full cert: http://wiki.cacert.org/ConvertingPgpKeyToCertificate

Thank you all for your quick reactions !!

To summarize if I want to use the PDF built-in signing I will need to convert my OpenPGP into a X.509 cert otherwise I can simply use the OpenPGP file signing

I want to stick to the UNIX Philosophy especially:

Write programs that do one thing and do it well.

So I will use the OpenPGP signing tool :)

Thanks !