I found this co-working space BASES (ベイシズ), which is conveniently located on the right in front of Hakata Station Chikushi (or Shinkansen) Exit.
They have visitor's fee at the reasonably priced 1,080 yen per person for any number of hours from 9 AM to 5 PM, and 1,500 yen from 5 PM to 10 PM (only on weekdays; weekend and national holidays closed.)
It's spacious space, very clean and comfortable place to work. One can concentrate far better than any Starbucks or coffee shops in a busy station or on a bustling street. In the premise, they even have a free-drink vending machine and also a "Lunch Map" indicating all restaurants and eateries around the building (and needless to say, foodies' paradise Hakata Station is in the proximity.) I went to a convenient store Circle K Sunkus (サークルKサンクス) on the same street of the building to get my cozy lunch.
Moreover, the most memorable moment for me was that I wore my ID card (for the visitor) after four-year-long to go out and come in the premise and could pretend that I was one of the "office workers" in the area. During my stay, I noticed a few foreigners (English and Chinese speaking people) discussing something in the open area. They have a "Concentration Area" for those who do not want any disturbance.
3F, Office New Gaia Hakata Eki-mae,
1-12-17, Hakata-eki Higashi, Hakata-ku,
Fukuoka-shi, Fukuoka, 812-0013
Weekdays: 9:00 - 22:00, Weekends / holidays: 9:00 - 17:00