Your bike has a seatpost, or at least what's left of one. That stub sticking up above the seatpost clamp is the remains of a conventional seatpost minus the saddle clamp. Loosen the clamp bolt and try to remove the stub. Hopefully it isn't corroded in place too badly. Use a pipe wrench or Vise-Grips to try to rotate it and pull it out. If it is badly stuck do a search for "stuck seatposts" and you will find a lot of possible removal techniques.

You do not have a "quill" seatpost or anything else, just a normal style and when/if you remove the stub, the proper diameter should be engraved on it.