Your existing ceiling box probably won't work for a fan. You could mount the fan up there, but in a year or two, one day you would be using it and it will fall out of the ceiling. The weight and vibration will just pull the box loose eventually.
Plus you would have a cord running over your fan blades. Not a good idea.
Personally the way I would do it is with Wiremold.
It's available from your local Lowes or HD. Get a surface mount outlet box which will drop your outlet for the garage door down an inch or two. It has cutouts on the side for raceway, so run one over to a good spot for the fan. Mount the round wiremold box for a fan directly to a joist for strength. Wire it up and mount the fan. Fairly easy. Looks nice. And it eliminates the problems with cords near fans, and with a fan mounted to a light duty box.