Short answer: That would be cool, but nope!
Cool and interesting answer: No, but the opposite actually did happen! See final paragraph below for further details.
(Oops this actually became the SUPER) long answer:
To start at the beginning, the United States Constitution was the second attempt at a foundational document establishing the governing structures of the US. The first was the Articles of Confederation, which was largely influence purely by fears of the what the British monarchy had enacted on the colonies. For example, the newly independent US was terrified of having a strong central government because the last thing that they wanted to have was a king. So, the Articles of Confederation didn't even grant the federal government the right to levy taxes or issue a uniform national currency. This lead to a massively weak and ineffective national government, so the next attempt was the United States Constitution.
The major influences on the Constitution were reactions to the British tyranny that had initially sparked the American Revolutionary War, reactions to inefficacy of the Articles of Confederation, and Englightenment ideals. Starting with the first aspect, similarly to the initial attempt, the Framers were hesitant to invest too much power in the federal government and especially too much power in a single person. This is where we get the system of "Checks and Balances", such as the President being able to veto an act of Congress, Congress being able to override the Presidential veto, and so forth. This meant that no single person could consolidate enough power to replicate what the Americans had fled from under the former monarchy. Another instance of this influence is the adoption of the Bill of Rights; The British army had forced American colonists to feed and house them, thus the new Constitution specified (in Amendment three) that no citizen would be forced to house soldiers. Next, the "second attempt" responded failed first attempt. Main issues in the Articles of Confederation were simply that the federal government had little to no actual power or ability to regulate the country, so the new Constitution gained the powers to create a national military, wage taxes, print and regulate currency, and so on.
By far the largest influences on the creation of the US Constitution-- that also in fact informed the earlier influences of objections to monarchy and the question of how to construct a functioning but limited government-- was Enlightenment ideals.
Briefly, I'd like to address the issue of asking about "Native American governing structure." Obviously this was well intentioned, but it still seems necessary to say that there was no united Native American government, there were more than 200 independent groups that we could refer to as Native American, and many represented wildly different structures of government.
However, although they had no influence on the structure of the US government, so particularly fascinating ideas of governance (in a wider interpretation of the word) in America include: Pueblos in Northern New Mexico and the Shosone and Ute in Utah. Just of particular personal interest.
The reason that Native American groups would not have influenced the US Constitution is primarily that first, the new US government was largely hostile to Native tribes, and two, the Framers were the uppermost social group of highly educated white men, and would not have considered the thinking or ways of life of native groups to be legitimate or worthy of consideration in the creation of their own state.
But! The reverse did happen: the US Constitution actually did influence Native American group's governing structures. After the United States extended it's territory across the continent, and finally decided to officially recognize the semiautonomous nature of tribal government, new organized tribal government emerged across the US. Most of them actually adopted te=heir own Constitutions and guaranteed rights to their citizens. For example, the Because of the 1937 Indian Reorganization Act, the Utes (group in what is now Utah) formed a government with a constitution and a tribal council system, incorporating both US ideas and Ute-originated ones.
As /u/cnzmur mentioned, it's usually the Haudenosaunee (the Iroquois Confederacy) that gets credit for having an influence on the founding fathers of the United States. On one hand, I generally feel that this influence is overstated. There are plenty of European and Euroamerican precedents for the Constitution, such as the United Colonies of New England, and the founding fathers didn't include some of the more unique aspects from the Gayanashagowa (the Iroquoian constitution) that would emphasize the connection.
But for those who favor the idea that the founding fathers were influenced by Iroquoian ideas in a significant way, there are two main lines of reasoning, which I discuss in these two posts: