As other comments noted, the quote is attributed to LBJ in Ronald Kessler's book, and was supposedly said to two southern governors. But in the absence of a reliable objective record of that quotation, among the best sources to answer your question are the presidential recordings made during the Johnson administration, which I've listened to at length during my undergrad studies. Several hundred conversations were recorded dealing with issues of racial politics and the Civil Rights Act of 1964. A discussion of those conversations can be found here.

Now, a quote like the one in OP's question is not found in any of these recordings, nor do they contain the oft-cited statement to Bill Moyers that LBJ had "lost to the South to the GOP for a generation." But they do provide excellent insight into how LBJ talked about these issues in private. For example, it is simply undisputed that LBJ did use the prevailing southern racial slurs of the time, including the word "n-----r," as this actual recording demonstrates. This is hardly the only example.

Another famous example is cited in "Lone Star Rising: Lyndon Johnson and His Times, 1908-1960", by Robert Dallek. Johnson defended the Supreme Court appointment of the famous Thurgood Marshall, rather than a black judge less identified with the civil rights cause, by saying to a staff member, "Son, when I appoint a n-----r to the court, I want everyone to know he's a n-----r."

That being said, for a rural-born white Texan in the late 1960s, the collected recordings show that LBJ had some astonishingly progressive views on race in America, but his nomenclature leaves something to be desired. It is also worth noting that LBJ knew his audience, and would speak differently to a Georgia state legislator than, say, a Connecticut governor. It's very difficult to tell when LBJ is putting on an act for audience or when he's speaking with his "true" voice. Additionally, I tend to detect a bit of self-aware irony in some of LBJ's discussion of these issues. I think that's key to understanding how LBJ could say the most radically progressive statements while simultaneously using a racial slur.

But given the tone and tenor of LBJ's conversations on these issues, I think the best we can say is that the quote cited by OP is not inconsistent with LBJ's style. But it seems unlikely that we will ever know if those exact words were uttered. From my own studies, I personally think the quote is either genuine or a fair paraphrase. It's the kind of thing LBJ might say to a Dixiecrat to convince them not to oppose the CRA. Thus, if anyone got "tricked" over the CRA, it wasn't black America -- it was Southern conservative democrats. In other words, as u/Thurgood_Marshall and others note below, while the quote might be genuine, the sentiment was not.

So, I think it does a disservice when these kinds of quotes are used to suggest that LBJ was duplicitous and uncaring about black America. Personally, I have no doubt over the bona fides of LBJ's empathy and humanity after listening to the conversations in June of 1963 during the Freedom Summer disappearances, where LBJ was positively distraught.

It's also revealing that on July 24, 1964, Barry Goldwater met with LBJ in the White House to discuss the issue of race in the presidential campaigns. Goldwater suggested that both campaigns not address the issue of race in order to avoid inflaming racial tensions. LBJ strongly disagreed. There is no doubt from the presidential recordings that LBJ considered race problems to be an issue of morality, going so far as to compare the CRA of 1964 as a moral challenge similar to that facing the Lincoln administration over whether to pursue the 13th Amendment.

EDIT: Because it hasn't been mentioned yet, I will add that the source of the quotation, Ronald Kessler, is not what I would call an unbiased historian. Mr. Kessler has been the chief Washington correspondent for the far-right-wing Newsmax. He is known for incorrectly reporting that Barack Obama was present for Rev. Wright's "white arrogance" sermon, among other controversies. I'm not saying he's a bad journalist, and I'm not indicting his politics. I'm just saying that he comes to the table with his own biases.

TOP LEVEL EDIT: The more I consider the OP's quote in light of my own time listening to the recordings at the LBJ library at the University of Texas, the more I am convinced that LBJ is putting on an act to these two southern governors to quiet their rancor over his pursuit of the CRA. I think the statements in OP's quote directly conflict with Johnson's private understanding of just how transformative the CRA would be, both politically and legally.

Also, I think he absolutely knows, at the moment that he is smiling and assuring these good ole boys that their political future is secure, that he is actually about to end their careers, as well as the present and future careers of a great many other conservative democrats in the south (as reflected in the Moyers quote, above).

I won't begin to deny that LBJ was a ruthless political operator, and indeed, an unrepentant liar. But I cringe that this single quote, robbed of its context, would be used by some to imply that LBJ was a heartless racist manipulator. That's one notion that I think the historical record soundly disproves.

GOLD EDIT: Wow, much thanks. I can't believe my first gold post is littered with n-bombs.