The New York Times does not use sales numbers alone, and will remove books if they suspect they are inflating their numbers. However, there is no evidence that is the case here. It simply does not appear to have the sales.
There is no reliable source for the claim. Other book lists do not place the book at #1. Alex Jones's other claims about his book's sales are absurd. Both men are habitual liars.
The sources are habitual liars
Let's start with the sources of the claim. Jones and Carlson are prone to extreme hyperbole, spreading conspiracy theories, and outright lies.
Alex Jones has been dragged through court for lying and promoting conspiracies numerous times for profit, and still is as of this writing.
Fox News has gone to court to defend Tucker Carlson's lying, arguing that he should not be taken literally. In McDougal v. Fox News Network, LLC, the judge concluded...
Fox persuasively argues... that given Mr. Carlson’s reputation, any reasonable viewer “arrive[s] with an appropriate amount of skepticism” about the statements he makes... Whether the Court frames Mr. Carlson’s statements as “exaggeration,” “non-literal commentary,” or simply bloviating for his audience, the conclusion remains the same—the statements are not actionable
Tucker Carlson is also a self-professed liar.
“Well, it’s—I guess I would ask myself, like, I mean I lie if I’m really cornered or something,” Carlson said. “I lie. I really try not to. I try never to lie on TV. I just don’t—I don’t like lying. I certainly do it, you know, out of weakness or whatever.”
Carlson is also not an impartial reporter, having this to say about the book.
If Alex Jones is just a crackpot, why are the most powerful people in the country trying to silence him? No one bothers to censor the flat-earthers. Maybe Alex Jones is onto something. Read this book and decide for yourself who’s crazy.
Jones was recently pushing an idea that soda can liners are part of a conspiracy to shrink penises.
We can rule them out as reliable sources.
Alex Jones's Full Claims, And Other List's Rankings
The New York Times is a constant, and usually unfounded, target of conservative ire about "fairness" and "bias". Their claim ignores that other lists also did not rank Jones's book at #1 making it either selective to build on existing skepticism of the Times, which they prmote, or another broad complaint about "the media", which they also promote. Or both.
So it is the number one book, not just fiction, not just non-fiction — all of them. More than any textbook, more than any Harry Potter book...
Publisher's Weekly puts the book in "New & Notable" ranked at "#2 Hardcover Nonfiction, #11 overall". Saying it sold "sold more than 30K copies in its first week". This put Jones's book at less than half the sales of #1 "It Ends With Us" at 69,619. They put Jones's book at #2 in "hardcover non-fiction".
As of Sept 13th, 2022, the day before Alex Jones made his claim Amazon listed it at #13 in Books, and #1 in specific book categories. As of this writing it is #7, possibly pumped up by the outrage generated by Tucker and Jones.
"Harry Potter and the Half-Blood Prince" sold 6,900,000 copies in the first day. TheAsh points out in the comments that Jones may have been talking about Harry Potter's current sales. If so, selling better than a 15+ year old book series isn't much of a boast, but because Harry Potter is so popular it might seem so to a non-critical listener.
New York Times's List Is Not A Simple Ranking Of Book Sales
The three bestseller lists mentioned in this answer use different methodologies. Publisher's Weekly is, presumably, getting their numbers direct from the publishers. Amazon is, presumably, getting them from their own sales. The New York Times polls book vendors. NYT's "About the Best Sellers" says...
Rankings reflect unit sales reported on a confidential basis by vendors offering a wide range of general interest titles published in the United States. Every week, thousands of diverse selling locations report their actual sales on hundreds of thousands of individual titles. The panel of reporting retailers is comprehensive and reflects sales in tens of thousands of stores of all sizes and demographics across the United States.
The Times will also remove books if they suspect the sales have been manipulated. Authors and organizations will make bulk buys of books to artificially inflate their sales numbers. Ted Cruz's "A Time For Truth", "Handbook for Mortals", and Donald Trump Jr.'s "Triggered" were found to have been manipulated and removed from the list. The NYT states...
Sales are defined as completed transactions by vendors and individual end users during the period on or after the official publication date of a title. Institutional, special interest, group or bulk purchases, if and when they are included, are at the discretion of The New York Times Best-Seller List Desk editors based on standards for inclusion that encompass proprietary vetting and audit protocols, corroborative reporting and other statistical determinations. When included, such bulk purchases appear with a dagger (†).
And the vendors must allow their numbers to be verified.
Publishers and vendors of all ranked titles must conform in a timely fashion to The New York Times Best-Seller Lists requirement to allow for examination and independent corroboration of their reported sales for that week.
The New York Times has acknowledged, on numerous occasions, that their bestseller list is not strictly based on sales. They consider it an editorial and they have gone to court over this in Blatty vs New York Times Company (1986).