John mentions the eight days on two different occasions. The two times are in consecutive paragraphs, so once you know the general location in the story, it is fairly easy to find the "eight day" information. The first mention of the eight-day journey can be found in the paragraph following the part of the story that has John shooting and killing the panther. John tells readers the following piece of information.
It is eight suns' journey to the east and a man passes by many Dead Places.
"Eight suns' journey" is presumably counting the number of times the sun is out—sunrise to sunset. John hasn't completed the journey yet, and the eight days' count is confirmed in the following paragraph when John finally arrives at the banks of the great river.
Toward the setting of the eighth sun, I came to the banks of the great river.
Notice that the sun is setting, and that confirms that it takes eight full days for John to go from his home to the Place of the Gods.
Then I raised my eyes and looked south. It was there, the Place of the Gods.