In the case of ancient Egypt, the answer is yes. Their mythology and history expressly stated that they were the first civilization. They believed that the universe was created on a "mound of creation" (as it's called by Egyptologists), which rose out of the receding flood waters of the Nile. There are various versions of the creation story, but in the most exciting one Atum or Re-Atum created the first pantheon of deities by masturbating them into existence: "Atum is the one who came into being as one who came (with penis) extended in Heliopolis. He put his penis in his fist so that he might make orgasm with it, and the two twins were born, Shu and Tefnut."^1 This mound of creation is called Benben by Egyptologists (Egyptian 𓃀𓈖𓃀𓈖𓉴), and it was a common feature of temples for millennia of Egyptian history, demonstrating the incredible longevity of this idea.^2
Most cultures have a creation myth that puts them at the center, so that doesn't completely answer your question. In Egypt, however, they kept written records of their history, which referenced their understanding of creation. There are frequent allusions to the "original time" (𓅮𓏏𓏖) in Egyptian texts, including the Pyramid Texts, and a glimpse of what they understood this to mean is found in the Turin Royal Canon.^3 That document records the reigns of kings from the New Kingdom back to the reigns of well-known Egyptian deities, including Seth, Horus, and Thoth. The tomb of the First Dynasty king Djer (also in the Turin Canon, II.13 in the facsimile) was later venerated as the tomb of the god Osiris.
So not only did they believe that they were the first civilization, created ex nihilo by the gods, they had places where they could go and see those gods' tombs in person. Their official history recorded the descent of their living kings from ancient deities, and they continually referred to these deities in contemporary texts. When they found old things, they knew that those things belonged to their civilization, and that they were directly descended from the people who made them.
Sources:
Translation from: Allen, James P. 2015. The Ancient Egyptian Pyramid Texts. See the original texts here. This quote is in PT IV (422-538), p. 279.
Obviously there is more to be said here. Representations of the mound of creation changed over time, but the central idea was always present. There's a great deal on this in: Kemp, Barry J. 2005. Ancient Egypt: Anatomy of a Civilization.
The Wikipedia page is actually a great source for this. It has the original text, a transcription, and translation.
Edit: Thanks for the 𓋞!
This is an answer I gave to a similar question a few years ago.
So I'm only gonna be talking about the Sumerians here. The only way that we are going to have any knowledge of what the first civilizations knew of their past is through written records. Writing first appears in Sumer in around 3000 BC, this is during the Uruk period (4000-3001 BC). At this time it was very much proto-writing and did not develop into a full written language until the Early Dynastic Period (2800-2500 BC)
It is not until the end of the Early Dynastic Period (2300 BC) that we see the emergence of the Sumerian King List which documents the kings of Sumer and gives us an insight into what the Sumerians may have believed about their past.
The King List, a collection of several sources, details the rulers of the cities of Sumer from the first antediluvian rulers to the last dynasty of Isin. The first king was Alulim, first king of Eridu. He ruled for 28,800 years. Now at first glance that may seem an unreasonably long rule and that is because it is. Alulim, the first king after the kingship descended from heaven created by the god Enki is clearly almost entirely mythical. This list of antediluvian kings ends with Ubara-tutu and the coming of the great flood that wipes the world clean.
The list resumes with Jushur the first of the dynasty of Kish. He ruled for 1200 years. 20 kings and 15255 years later we have En-me-barage-si who is the first king we have archaeological evidence for. He is dated from around 2600 BC from two pieces of alabaster vases found at Nippur which bear his name. Though he ruled for 900 years, a rather long time, it can be surmised that he is indeed real. He is mentioned also in the Epic of Gilgamesh alongside Gilgamesh himself giving credence to the thought that Gilgamesh is a historical figure. The King List continues into the time of rulers that can easily be verified such as Sargon of Akkad who ruled for 40 years and founded the Akkadian Empire. Thus we see the transition of the mythical into the semi-mythical and then verifiable history. But more on that later.
The Sumerian creation myth is important to note in this discussion. It recounts that the gods Enki among them created the first “black-headed people” (the Sumerians) and settled them in the land giving them the kingship and thus the first cities were created. A large part of the story is missing but at some point the gods decide not to save mankind from a flood which strikes destroying man and cites. Later the world is presumably repopulated.
In addition there is the “Debate between Summer and Winter” a creation myth from the mid 3nd millennium. This details the creation of the land and seasons by Enlil. In it he is seen to irrigate the land “guaranteeing the spring floods at the quay” and to begin the agricultural tradition of the land “making flax grow and barley proliferate.”
Finally and most interestingly for this topic is the “Debate between Sheep and Grain” another creation myth written in the mid-3rd millennium . The myth details a time in which sheep and grain were unknown to the land. The people “went about with naked limbs in the Land. Like sheep they ate grass with their mouths and drank water from the ditches.” The myth ends with the virtues of grain being extolled “from sunrise to sunset may the name of Grain be praised. People should submit to the yoke of grain.”
Therefore we can see that early history of Mesopotamia, the Ubaid period and before, is in Sumerian text seen in a divine light. The land was created by the gods as were the people and they were given cities and kingship. Enlil gave the people the summer and the winter, he gave them wheat and irrigation as Enki gave them kingship. Only in the “Debate between Sheep and Grain” is there indicated any knowledge of a time before sedentary agriculture. This myth clashes with that of the “Debate between Summer and Winter” though it is part of the same tradition indicating the lack of a unified view of their past. It seems that the Sumerians saw their past as part of a very real mythical tradition. Their kings begin as mythical figures and progress towards the non-mythical. The mythical and the non-mythical are closely linked in the Sumerian view of themselves and their past
I would conclude that the Sumerians did believe themselves to be not just the first civilization but the first people, it is part of their creation myth. In addition there was no knowledge in the way we would think of a hunter-gatherer life preceding their urban civilization. If there is any hint it exists as another facet of the extensive and contradictory creation myth of the peoples of Sumer.
I answered almost this exact question a couple months back, in this thread here. Which includes links to a number of sources, but the short answer is that cities arose gradually and people didn't stop being nomadic hunter gatherers all at once. It was a slow transition that often looked like a hybrid between settled urban agricultural life and hunting and gathering. The evidence we have can't tell us what exactly they thought about the change but the safest assumption is that the transition was gradual enough that it wasn't seen as too drastic of a departure at the time.
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Follow up/related question: Are there any 'meta' early writings, acknowledging the fact they're an early writing (in any capacity)?
