Here's a good example from "Traces on the Rhodian Shore":
The idea that there is a continuous interaction between man and his environment—man changing it and being influenced by it—also has its mythological antecedents, but its full development belongs basically, I think, to rational thought, because such a conception requires a sense of history. The Sumerian thought the civilization of which he was a part—its institutions, cities, towns, farms, and so on—had been more or less the same from the beginning
And from Kramer's "Sumerian Historiography":
That Sumer had once been desolate marshland with but few scattered settlements, (...) —such thoughts probably never occurred even to the most learned of the Sumerian sages.
I must admit I am very skeptical of this statement. Obviously, the ancient world did not have the rapid-fire change we're experiencing now, but surely cities rose and fell, environmental disasters must have displaced people, foreigners introduced new pantheons and people would have a sense lots of stuff happened within their lifetimes.
So which is it?
This is a very interesting subject, but historical consciousness is indeed a relatively recent innovation.
Of course, people have always felt the passing of time and realized that things change. Still, they placed this within much broader cycles where a single lifetime was relatively meaningless. Barbara Adam actually mentioned the Sumerian king lists are an excellent example of this, with the earliest kings being noted as having reigned thousands of years. The same is true of one of the earliest passages in the Bible, where Abraham's descendants are all said to have lived extraordinarily long lives. According to Adam, there was a tendency among the earliest civilizations to place their prehistory in a world before time (think of the Garden of Eden), and their current state of mortality as a fall from that world. We could also add the whole cyclical time thing, though that's a whole can of worms of its own. What it comes down to though is that even something which could be considered a work of history, namely the king lists, are actually themselves mythologized depictions of the past.
Now you could argue that for example the Greeks did write actual historiography. I will quote Zachary Schiffman on this: "Classical historians, however, did not perceive these [historical] differences. Instead of 'the' past -- whatever kind of entity it might encompass -- they conceived of multiple 'pasts' characterized by different time frames, each disassociated from the next, without privileging any particular one." (Schiffman p. 5) According to him, this is caused by a lack of the sense of anachronism. This was anything but unique to the Greeks, another interesting case study would be the Chinese. Historiography played a major role in premodern China, but it had a ceremonial function. The goal of historians wasn't necessarily to think about what happened in the past, but to record what happened in the present. Just as Eden, the goal was in a sense to compare the current state to "an ideal society once realised in the past Golden Age." (Schneider p. 239) There was no historical progress, only constant cycles.
The same was true in medieval Europe, which can be illustrated by talking about their chronicles. Sure, they had the linear time people so often like to associate with modernity. Yet, just as the Chinese, the goal was at times to record the present for posterity's sake more than it was to learn from the past. They were composed chronologically, with no distinction being made between for example a new church being built or the death of Charlemagne. The medieval worldview didn't leave much room for men to make history, instead the world had been ordered by God and 'historians' only had the job of recording this. In a sense you could link the rise of history to the gradual death of God as modernity approached. Then again, I'm just being purely speculative when I say that.
So when did history start? For Schiffman the origin of our idea of anachronism, of historical consciousness, derives from the Renaissance. The discovery of Latin texts that were wildly different from medieval Latin led to an increased attention to the historical process of change and the distance that had emerged between past and present. There's a strong link to be made as well to the impact of philology in this regard. Reinhart Kosseleck would probably place it during the Industrial Revolution. His ideas about the creation of the past and the future remain influential today, but to get into it here would take me too far (though Assmann's work is partly inspired by him if you're interested). Aleida Assmann linked the birth of history directly to the French Revolution. In a fell swoop, the Ancien Régime had become history, something which can be taken quite literally in this context. Assmann stated that one of the immediate reactions to this was to start up institutions tasked with preserving the relics of this sudden past and study them as 'sources'.
Debates about this very subject are of course ongoing but I dare to state that most historians would agree that the historical consciousness is a modern phenomenon. The specifics are still up for grabs though, meaning not only when exactly a historical consciousness arose, but also how to understand premodern times exactly. Of course I've skipped through a lot, meaning there's still plenty to be said about this. That said, I hope my jump through time and space gave you a first glimpse into quite an interesting subject that cuts right at the core of what history really is.
Further reading:
Assmann A., Transformations of the Modern Time Regime. In: Lorenz, C. & B. Bevernage, eds. Breaking up Time: Negotiating the Borders between Present, Past and Future. Göttingen: Vandenhoeck & Ruprecht, 2013, pp. 37-56.
Schneider A., Temporal Hierarchies and Moral Leadership: China’s Engagement with Modern Views of History. In: Lorenz, C. & B. Bevernage, eds. Breaking up Time: Negotiating the Borders between Present, Past and Future. Göttingen: Vandenhoeck & Ruprecht, 2013, pp. 236-251.
Schiffman Z. S., The Birth of the Past. Baltimore: The Johns Hopkins Press, 2011.
Adam, Barbara. Time. Oxford: Polity Press, 2004. (this is not necessarily about historical consciousness, but the early chapters are a nice introduction into premodern time perception)