The invite is the quest.

Region: Jeuno, Lower
Receive At: Inside Neptune's Spire Inn, I-7
Receive From: Ghebi Damoohe
Minimum Fame Level: 1
Requirements: None
Previous Quests Needed: None
Reward: Tenshodo Member's Card, Tenshodo Invite
Title Given: Tenshodo Member
Repeatable: No

How to complete: When you talk with Ghebi Damomohe there will be an option box that appears with two visible options and a third hidden on at the bottom. Select this hidden one to start the quest. Ghebi will tell you that if you wish to join the Tenshodo you will either need an Tenshodo Invite from a current member or you must get an application in Bastok. You can either buy the invite from the Auction House, get one from someone you know, or head to Port Bastok and get your own application from Silver Ole in Warehouse 2 at F-6. Once you have the application then head back and talk with Ghebi Damomohe to become a member of the Tenshodo.

***EDIT***

Info obtain'd from http://www.ffinfo.com/ffxi/quests/de...o%20Membership

But somepage or allakhazam are probly better sources