It appears the scene that /u/silas_of_the_lambs is referring to is in the opening seconds of this video. There's not much context there, other than the exchange occurring as the U.S. Navy was preparing to execute its carrier raids on the Marshall and Gilbert Islands that were carried out on February 1, 1942.

For readers less familiar with these individuals, let me introduce them briefly. Admiral Chester Nimitz was the newly-appointed Commander in Chief of the Pacific Fleet at the time (CINCPAC). Layton had been the Combat Intelligence Officer on the staff of Nimitz's predecessor, Admiral Husband Kimmel, before and during Pearl Harbor. Nimitz retained Layton in that position and kept most of Kimmel's staff when he arrived in Hawaii. Admiral Isoroku Yamamoto was the Commander In Chief of Japan's Combined Fleet at the time. He was not the highest-ranking naval officer in Japan, but he oversaw all fleet operations (as opposed to serving in a purely administrative or political role). Admiral Chuichi Nagumo was in command of Japan's First Air Fleet, also known as the "Kido Butai," which consisted of Japan's six fleet aircraft carriers and its supporting vessels.

As you've correctly identified, the exchange between Nimitz and Lieutenant Commander Edwin Layton seems to have been modified. In chapter 26 of his memoir, And I Was There, Layton recalls a conversation he had with Nimitz at about this time:

"I want you to be the Admiral Nagumo of my staff," Nimitz told me. "I want your every thought, every instinct as you believe Admiral Nagumo might have them. You are to see the war, their operations, their aims, from the Japanese viewpoint and keep me advised what you are thinking about, what you are doing, and what purpose, what strategy motivates your operations. If you can do this, you will give me the kind of information needed to win this war."

It's worth pointing out that Layton's book was published in 1985, one year after his death and more than 40 years after the events in question. Even if he had written the book years before, it seems possible that this is more of a recollection of a conversation or series of conversations with Nimitz rather than a word-for-word quote.

Despite this interaction with Nimitz, Layton's book is scant on information relating to Nagumo. I can find no indication that Nagumo and Layton ever met or had any personal interaction. However, Layton and Yamamoto certainly did know one another.

Layton was assigned to Japan twice before the war. Early in his career, Layton met a group of Japanese naval officers whose ship was visiting the West Coast and noted that nearly every Japanese officer spoke English or French, while neither Layton nor his fellow officers spoke any Japanese. Layton requested to be sent to Japan for language training and was there from 1929-32. He later returned to Japan as a naval attache at the American embassy in Tokyo from 1937-39. Yamamoto had spent two stints in the United States earlier in his career, including one as naval attache at the embassy in Washington. Yamamoto learned English and developed relationships with many American naval officers during that time, but he and Layton met in Japan.

The relationship between Layton and Yamamoto (as well as Yamamoto and other American officers) was certainly professional, but it also included more unofficial interactions. Yamamoto invited Layton to a kabuki theater show that they attended together, and later to a duck hunt with several Japanese officers. They played bridge together in Tokyo, and Yamamoto had developed a reputation as a card shark at American poker games in Washington when he was there. Yamamoto was well-liked by American officers and respected as a professional. Layton's book is sympathetic toward Yamamoto, portraying him as a commander who respected the Americans and was reluctant to go to war, but served Japan as best as he could.

Layton's respect for the Japanese admiral was of the factors that led to Yamamoto's death. U.S. Navy signals intelligence intercepted a radio message including information about Yamamoto's travel schedule through the Solomon Islands in early 1943. Layton was the one who suggested to Nimitz a special mission to shoot down Yamamoto's plane. Layton recommended the assassination mission, believing that Japan had no other naval officers who were as capable as Yamamoto.

From Layton's book:

There were some qualms of conscience on my part. I was signing the death warrant of a man whom I knew personally. It was impossible for me not to feel for Admiral Yamamoto with a certain amount of fondness. It was not as though we were involved in the shooting of somebody whom I had never met. Although he had been my "official friend" four years ago, however, I could not dismiss the fact that he was now my sworn enemy. War, in essence, involves the destruction of your enemy. As the leader and inspiration for the Japanese navy, Yamamoto was an important foe whose death would benefit the Allied cause.

Here is another retelling of the story from Layton to U.S. Naval Institute. The full text also includes some more information on the pre-war relationship between Layton and Yamamoto.

(Yamamoto) was the outstanding symbol of Japan’s navy and the Pearl Harbor surprise attack and the only man in the Japanese Navy who I thought was very superior. This may have prompted Admiral Nimitz to ask, “Are you sure there are none who can take his place and be better than him?” I said: “Absolutely none. Absolutely none.”