You can easily store your luggage at Fukuoka airport.
Left Luggage Service
Sagawa offers manned luggage storage daily from 7.30 a.m. to 9.40 p.m. daily. Its counter is located in the international terminal, just beside arrivals on the first (ground) floor. Daily charge for small/medium/large consignment is 300/500/800 yen. Sagawa, as well as other companies in the terminal, can also offer onward transportation of luggage.
As in many places in Japan coin lockers are available (we reviewed such lockers generally here). There are lockers in both domestic and international terminals.
These lockers come in three sizes and are plentiful. They are accessible only when the terminal building is open: 7 a.m. to 10 p.m. for the international terminal and 7 a.m. to 9 p.m. for the domestic terminal. Be aware that calendar days are used to determine cost so e.g. using one from eight o’clock at night to seven o’clock the following morning may only be eleven hours, but will be charged as two days.
The daily locker rental cost is 300/400/500/700/800 yen based on size.