The simple answer is no. The historical fallacy at the heart of this question is that Japan surrendered because the USA dropped two nuclear bombs on Japanese cities.
It is considered that the real reason for Japan's surrender was the declaration of war against Japan by the Soviet Union. Japan was well aware that Russia had a score to settle with them over the Russian defeat by the Japanese in the early 1900's. Japan was driven close to defeat fighting the Allies in the Pacific, so the opening up of a 'Western Front' fighting Russian forces would have led to inevitable defeat.
The majority of Japanese cities had already been destroyed by American bombing. In fact, it was difficult for the Americans to select target cities for nuclear bombing. They wanted to make it clear that nuclear weapons had been solely responsible for destroying the targets. previous bombing raids on a city might have left behind, for example, time bombs to annihilate the target.
There is also the distinct possibility that the use of the atomic bombs were used to demonstrate to the Soviet Union that the USA possessed a super-weapon. This was intended to shape global politics in the post-war era.
If Japan had its nuclear weapons it would have been trapped between two massively powerful adversaries. Namely, the USA and the USSR. Even the USA had a limited number of atomic bombs. Often the estimate is three. These would have been used up with the Trinity test and the bombs used on Hiroshima and Nagasaki. In the OP's alternative history scenario the USA could have easily had many more. (Indeed, it is conceptually possible that any alternative history where there are more nuclear weapons at the end of the Second World War the USSR could also have its own nuclear weapons too.)
Decision makers in Japan would be faced with determining how they could use their nuclear weapons most effectively. This means against the USA and the USSR. Even if only they are confronted with a nuclear-armed USA, they will still have to deal with Soviet conventional forces on a grand scale. Their best option seems to be surrender before they are engaged in fighting Soviet forces. This is essentially similar to the sequence of events in history as we know it.
What might be different is that if there are more nuclear weapons, is that more Japanese cities will be destroyed with nuclear weapons before Japan surrendered.
Please note this answer is based on historical research that demonstrates the Japanese surrender was more due to the Soviet declaration of war than the historical myth concerning the nuclear bombing of Hiroshima and Nagasaki.
@sphennings in comments requested links concerning the historical background. Admittedly the real historical research will be in books and history journals there are links to various aspects of the Japanese surrender.
From the Wikipedia entry on the Soviet-Japanese War (1945) illustrates the complexity of factors involved. To assume the surrender of Japan in 1945 was caused by the two atomic bombing is too simplistic. Simplification on this scale is the stuff of myth.
From the time of the first major Japanese military defeats in the
Pacific in the summer of 1942, the non-military leaders of Japan had
come to realise that the Japanese military campaign was economically
unsustainable — as Japan did not have the industrial capacity to
simultaneously fight the United States, China and the British
Commonwealth and Empire — and there were a number of initiatives to
negotiate a cessation of hostilities and the consolidation of Japanese
territorial and economic gains. Hence, elements of the non-military
leadership had first made the decision to surrender as early as 1943;
the major issue was the terms and conditions of surrender, not the
issue of surrender itself. For a variety of diverse reasons, none of
the initiatives were successful, the two major reasons being the
Soviet Union's deception and delaying tactics, and the attitudes of
the "Big Six", the powerful Japanese military leaders. (Refer to
Surrender of Japan for more detail.)
The Manchurian Strategic Offensive Operation, along with the atomic
bombings of Hiroshima and Nagasaki, combined to break the Japanese
political deadlock and force the Japanese leaders to accept the terms
of surrender demanded by allies.
In the "Sixty years after Hiroshima" issue of the Weekly Standard,
American historian Richard B. Frank points out that there are a number
of schools of thought with varying opinions of what caused the
Japanese to surrender. He describes what he calls the "traditionalist"
view, which asserts that the Japanese surrendered because the
Americans dropped the atomic bombs. He goes on to summarise other
points of view.
Tsuyoshi Hasegawa's research has led him to conclude that the atomic
bombings were not the principal reason for Japan's capitulation. He
argues that Japan's leaders were impacted more by the swift and
devastating Soviet victories on the mainland in the week following
Joseph Stalin's August 8 declaration of war because the Japanese
strategy to protect the home islands was designed to fend off a US
invasion from the South, and left virtually no spare troops to counter
a Soviet threat from the North. This, according to Hasegawa, amounted
to a "strategic bankruptcy" for the Japanese and forced their message
of surrender on August 15, 1945. Others with similar views
include The "Battlefield" series documentary, among others,
though all, including Hasegawa, state that the surrender was not due
to any single factor or single event.
A broader picture of the events leading to Japan's surrender can be found in the Wikipedia entry on the Surrender of Japan.
Foreignpolicy.com announced that "The Bomb Didn't Beat Japan ... Stalin Did"
The U.S. use of nuclear weapons against Japan during World War II has long been a subject of emotional debate. Initially, few questioned President Truman’s decision to drop two atomic bombs, on Hiroshima and Nagasaki. But, in 1965, historian Gar Alperovitz argued that, although the bombs did force an immediate end to the war, Japan’s leaders had wanted to surrender anyway and likely would have done so before the American invasion planned for Nov. 1. Their use was, therefore, unnecessary. Obviously, if the bombings weren’t necessary to win the war, then bombing Hiroshima and Nagasaki was wrong. In the 48 years since, many others have joined the fray: some echoing Alperovitz and denouncing the bombings, others rejoining hotly that the bombings were moral, necessary, and life-saving.
The following lecture on the Carnegie Council website further collaborated the historical role of the Soviet Union in Japan's surrender.
Look at the facts. The United States bombed 68 cities in the summer of
1945. If you graph the number of people killed in all 68 of those attacks, you imagine that Hiroshima is off the charts, because that’s
the way it’s usually presented. In fact, Hiroshima is second. Tokyo, a
conventional attack, is first in the number killed. If you graph the
number of square miles destroyed, Hiroshima is sixth. If you graph the
percentage of the city destroyed, Hiroshima is 17th.
Clearly, in terms of the end result—I’m not talking about the means,
but in terms of the outcome of the attack—Hiroshima was not
exceptional. It was not outside the parameters of attacks that had
been going on all summer long. Hiroshima was not militarily decisive.
The Soviet Union’s declaration of war, on the other hand,
fundamentally altered the strategic situation. Adding another great
power to the war created insoluble military problems for Japan’s
leaders. It might be possible to fight against one great power
attacking from one direction, but anyone could see that Japan couldn’t
defend against two great powers attacking from two different
directions at once.
The Soviet declaration of war was decisive; Hiroshima was not.
After Hiroshima, soldiers were still dug in in the beaches. They were
still ready to fight. They wanted to fight. There was one fewer city
behind them, but they had been losing cities all summer long, at the
rate of one every other day, on average. Hiroshima was not a decisive
military event. The Soviet entry into the war was.
Even Foxnews announced on 14 August 2010 that the Soviet Offensive was the key to the Japanese surrender was eclipsed by the A-bombs.
It was a momentous turn on the Pacific battleground of World War II,
yet one that would be largely eclipsed in the history books by the
atomic bombs dropped on Hiroshima and Nagasaki in the same week 65
years ago. But in recent years some historians have argued that the
Soviet action served as effectively as — or possibly more than — the
A-bombs in ending the war.
Essentially this information can be readily found by a Google search using the search terms "surrender Japan Soviet Union".