Do you mean automatically push new discussions to a social account owned by the forum admins, or add a button that allows regular users to share content they select to their own social account?
Since you mention social media icons I assume you are already using an extension. Are you referring to FriendsOfFlarum Share Social ? If Threads has a URL-based share API like the other social networks it shouldn't be difficult to add it to FoF Share Social.
For other kinds of automatic synchronisation/posting to other platforms, if you have your own developer, this is quite easy to do by using the PHP extenders and writing an event listener that then calls an external API with the required payload and credentials. No extension needed.