Quotations are often, as you are probably aware, invented or embellished into popular assumption. The specific formulation of the quote as

China? There lies a sleeping giant. Let him sleep, for when he wakes he will move the world.

Is unattested in contemporary sources. This blog post by Peter Hicks looks into the specific origins of the quote: its first claimed attestation in any language is supposedly in a piece by Lenin from 1923, while its first attestation in English seems to be in the 1963 film 55 Days in Peking, about the siege of the Beijing Legation Quarter during the 1900 Boxer Uprising. A general sentiment about a Chinese resurgence in European thought seems to date to the 1870s at the earliest.

That is not to say, however, that Napoleon did not comment on China. During his exile on St Helena, the second British embassy to China, that of Lord Amherst, got underway. Amherst set out from England at the beginning of 1816, arrived at Canton (well, technically, anchored off Hong Kong to avoid notice) in July, attempted (and failed) to attain an audience with the Jiaqing Emperor in September, and, after a shipwreck off what is now Indonesia, managed to return to England, calling at St. Helena in early April 1817. This may have prompted a discussion on China on 3 November 1816, of which we only know the existence, not the details, thanks to a brief reference in the work of the emperor's infamously sycophantic biographer, Emmanuel de las Cases.

However, Napoleon's doctor, Barry O'Meara, kept extensive journals of his time with the ex-emperor, and his entry for 26 March 1817 records comments by the two men on the Amherst embassy, whose members were due to call at the island in a few days. News had clearly been filtering in earlier, as Napoleon was evidently aware that Amherst had been happy to perform obeisance before the emperor (i.e. the 'kowtow'), but was cautioned against it by the 'Canton gentlemen' (chiefly the merchant George Staunton, explorer Thomas Manning and translator Robert Morrison) who had been brought on as advisors. This prompted this rather amusing line of discussion (some paragraph spacing added for the sake of the modern reader):

He said, he thought the English ministers had acted wrong in not having ordered him [Amherst] to comply with the customs of the place he was sent to, or otherwise not to have sent him at all. I observed, that the English would consider it as debasing to the nation, if Lord Amherst had consented to prostrate himself in the manner required. That if such a point was conceded, the Chinese would probably not be contented, and would require similar ceremonies to be performed as those insisted upon by the Japanese, and complied with so disgracefully by the Dutch. That, besides, Lord Amherst was willing to pay the same obeisance to the emperor as he would do to his own king.

Napoleon replied, "It is quite a different thing. One is a mere ceremony, performed by all the great men of the nation to their chief, and the other is a national degradation required of strangers, and of strangers only. It is my opinion, that whatever is the custom of a nation, and is practised by the first characters of that nation towards their chief, cannot degrade strangers who perform the same. Different nations have different customs. In England, you kiss the king's hand at court. Such a thing in France would be considered ridiculous, and the person who did it would be held up to public scorn; but still the French ambassador who performed it in England, would not be considered to nave degraded himself. In England, some hundred years back, the king was served kneeling, the same ceremony now takes place in Spain. In Italy, you kiss the pope's toe, yet it is not considered as a degradation. A man who goes into a country must comply with the ceremonies in use there, and it would have been no degradation whatever for Lord Amherst to have submitted to such ceremonies before the emperor of China, as are performed by the first mandarins of that empire. You say that he was willing to render such respect as was paid to his own king. You have no right to send a man to China to tell them that they must perform certain ceremonies, because such are practised in England. Suppose now, for the sake of example, that it were the custom in England, instead of kissing the king's hand, that he should offer his breech to be kissed by those who were presented to him; why then, forsooth, the Emperor of China must let down his breeches, –––––––, because it was the practice in England."

Napoleon went on to aggrandise himself, as was his wont, by asserting that his ambassadors, had he sent any, would most assuredly have performed obeisance before the emperor of the Great Qing, but then the topic turned to the subject of the implications of the Amherst mission's failure: had it jeopardised Britain's commercial interests, and could they be regained by war?

I said, that we could easily compel the Chinese to grant good terms by means of a few ships of war; that, for example, we could deprive them altogether of salt, by a few cruizers [sic] properly stationed.

Napoleon replied, "It would be the worst thing you have done for a number of years, to go to war with an immense empire like China, and possessing so many resources. You would doubtless, at first, succeed, take what vessels they have, and destroy their trade; but you would teach them their own strength. They would be compelled to adopt measures to defend themselves against you; they would consider, and say, 'we must try to make ourselves equal to this nation. Why should we suffer a people, so far away, to do as they please to us? We must build ships, we must put guns into them, we must render ourselves equal to them.' They would," continued the emperor, "get artificers and ship-builders from France and America, and even from London; they would build a fleet, and, in the course of time, defeat you."

Napoleon's comment is not, fundamentally, an assertion that China was a 'sleeping giant' that would dominate the world once it awoke. He was speaking, instead, in far more bounded and contingent terms about China's relative naval power to Britain, and the prospect of it already being able (potentially, anyway) to leverage its immense wealth to win over, if not state allies, at least private interests in the Atlantic world that would furnish it with the means to defeat Britain, specifically, in a conventional war, specifically. It is very possible to see how this quote could be retroactively read as a dire warning about the weight that an awakening China could throw about on the world stage, but at the time this was Napoleon discussing what he did best – military strategy – rather than what he arguably was less good at – international diplomacy.

The post I linked to earlier has other examples from O'Meara of discussions of China, from 27 May and 22 September 1817, respectively. The former comes in the context of further discussions regarding Lord Amherst, who had recently left:

"If," said he [Napoleon], "a million of francs had been given to the first mandarin, every thing would have been settled, and it would not have been a reproach to the nation; as that embassy was not one which regarded the honour of the country. It was, and ought to be considered more as an affair of merchandize than as one immediately affecting the nation. It was in fact an embassy to China from the tea-merchants in England, and therefore advantages might with great honour be purchased. Besides, when you send ambassadors to those barbarians, you must humour them and comply with their customs. They do not seek you. They never have sent ambassadors in return for yours, nor asked you to send any. Now great commercial advantages may be lost to England, and perhaps a war with China be the consequence. If I were an Englishman, I should esteem the man who advised a war with China to be the greatest enemy to my country in existence. You would in the end be beaten, and perhaps a revolution in India would follow."

He then turned, as one does, to Russia and the Ottomans:

" In the course of a few years," added he, "Russia will have Constantinople, the greatest part of Turkey, and all Greece. This I hold to be as certain as if it had already taken place. Almost all the cajoling and flattering which Alexander [of Russia] practised towards me was to gain my consent to effect this object. I would not consent, foreseeing that the equilibrium of Europe would be destroyed. In the natural course of things, in a few years Turkey must fall to Russia. The greatest part of her population are Greeks, who you may say are Russians..."

He later went on to say that although France and Britain (and Prussia) would likely form an alliance to attempt to prevent this, an Austro-Russian alliance would almost certainly prevail in a prospective conflict over Russian interests in the Ottoman Empire.