JR Kyushu Travel Agency

Is there a JR Kyushu Travel Agency in Fukuoka Airport where I can pick up a JR Pass and book seats?

Thank you.

There is no office there. You will have to go to the nearest JR station, which is Hakata Station. It's only one stop away, takes about five minutes. Here is the PDF map for Hakata Station: http://www.jrkyushu.co.jp/pdf/hakata_st_kounai.pdf

The pink writing with the arrow in the lower middle of the map is where the exchange office is.

But if you were looking to exchange the pass while waiting for a connecting flight at Fukuoka Airport, then I would suggest arriving at your final destination to do so. Chances are the city that you arrive in will have a JR station where you can exchange it.

There is no JR office, but you can buy it at the airport at the TISCO Travel Information Center on the 1st Floor Fukuoka Airport International Terminal.
See:
http://www.jrkyushu.co.jp/english/railpass/railpass.jsp
The only other places to buy it in Fukuoka City are the Hakata Port Beetle Terminal and Hakata Stn. That said, to exchange the voucher for the Pass, you'd have to do it at Hakata or Kokura Stn.
You could of course buy the e-ticket for the pass before you arrive in Japan. See the link above.
Fukuoka is a nice city to see though. Be sure to see the Nanzoin Temple with its Great Buddha. It is really impressive:
https://www.youtube.com/watch?v=lT4VQFmHFyM
For lots of info on what to see and do, see:
http://www.thejapanfaq.com/fukpic.html
https://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...
To get around the city, look into a subway day pass or Fukuoka Tourist City Pass
http://yokanavi.com/access_tourist_eg

Thank you very much

"JR Pass" is understood in this website to mean the national JR Pass which cannot be bought in Japan. Are you wanting to purchase a JR Kyushu Pass or the national JR Pass?