I want to generate an OpenPGP key and use it with Thunderbird, but I don't want this key to be synchronized with the keyservers when I do so in Seahorse.
No, there are no means of preventing keys from being uploaded to keyservers. Neither does RFC 4880 (OpenPGP) specify any restrictions, nor do the usual key server implementations support anything like "do-not-upload"-options.
The only chance would be to manually tell people not to upload a key; but I would not put expectations into such a desire being successful for a longer time span: users of OpenPGP are just used too much in keys being available on key servers, and at some point somebody will (accidentally) upload the key. And deleting from key servers is not possible to prevent deletion attacks and censorship.