It's been a few years since I read Chernow's bio of Grant and don't have it here. I don't recall if there was a specific Adams he didn't want to appoint. If it was Henry, it's possible to imagine various reasons. Henry started the North American Review with his brother after coming back from England (after serving as secretary to his father, who'd been the US ambassador there during the Civil War). Henry also did spend a good bit of his life expecting to be recognized for his great gifts and given a government position or at least a voise in it, like his friend John Hay. The North American Review had a scholarly side, but it also was a critic of the government and Grant's administration had a lot of scandals that made it a good target for it- and, depending on the date of your quote, Grant might have been annoyed at the criticism. The Adams family had also been in the business of US government for generations, and they were well-connected and superbly educated ...unlike Grant, who had a pretty basic education before West Point, and no family connections. Grant had no trouble appointing people who were well-connected and well-qualified ( Hamilton Fish , for example) but he might have figured that an Adams would come with an air of entitlement, ready to quote John Stuart Mill or La Rochefoucauld , or smugly recount past conversations with English lords to the cigar-smoking Army veteran from Missouri.

His specific antipathy to John Adams could be somewhat related. In 1782 the Founding Guys had left in place much of the old government- the colonial state legislatures- and gotten rid of the governors appointed by the king. There was very soon a question as to how to replace what had been a royally-appointed executive . Popular elections carried the risk of demagoguery and mob rule: but the 1787 Constitutional Convention essentially made the Presidency an elective office. John Adams, of all the Founding Guys, was the most uncomfortable with this. He admired the constitutional monarchy of Britain, and while he had no illusions about kings and hereditary aristocrats being recreated in the US, he had an elitist view, saw the general population as not very qualified to make wise decisions, and felt that a president should not be popularly elected. Adams, tellingly, even wanted the title to be much more glorious and elaborate than "Mr. President". As President, he himself acted sometimes almost as a monarch (as with the Alien and Sedition Acts). If Grant had read much of John Adams, he would have likely realized that John would have much preferred any of his many Harvard-educated, cultured great-grandsons for president, never would have considered a rather undistinguished self-made man like Grant suitable for the job.

Nagel, Paul C. ( 1999) Descent from Glory: Four Generations of the John Adams Family. Harvard Univ. Press.