Nothing on page 269 of this PDF says the word “Adams”, “J. Q. Adams”, “Q. A.”, or any variation thereof. To indicate certain authorship, to be clear. Your quote does not state “authored by”. I pressed CTRL + F - I searched what you provided, it isn’t there. It’s missing.
Neither does page 268. It isn’t there. It’s missing. Neither does page 267, which is the beginning page of that chapter. Neither does page 285, the last page of that chapter where you might expect to see footnotes.
The Table of Contents says nothing about guest contributions. There doesn’t seem to be anything at the back of the provided PDF book at all. It isn’t there, it isn’t anywhere. It’s missing.
If Adams did write this, this PDF does not appear to be offering any evidence at all that he did.
And like I said, I’d like to read it too. It is not a long work, judging by the variants available online. This kind of thing is exactly why I do readings, to enhance visibility. This is exactly the kind of item I’d like to get out there.
Allegedly JQA wrote it as an "unsigned" essay so I wouldn't expect his name there. Evidence of his authorship must come from elsewhere. I await delivery of the work allegedly claiming its JQA to see what evidence may exist there.