I cannot directly address the question, beyond saying "it depends on your source." But if you'd like context...

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Marx was not alone in the characterization. Adam Smith also maintained that there was a definitive change in the economic system. He did not go back to Ancient Greece, but he goes back to what we may call the end of Mercantilism. What I'm hoping is read into this is that a change did occur that Smith is attempting to explain. The merchants, in this case, are looking to get themselves rich. In his narrative they don't really understand the full economics, except that by pressuring their monarchs to alter the economic system they'll be rich. This happens to generally work out. Adam Smith:

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The different progress of opulence in different ages and nations has given occasion to two different systems of political œconomy with regard to enriching the people. The one may be called the system of commerce, the other that of agriculture. I shall endeavour to explain both as fully and distinctly as I can, and shall begin with the system of commerce. It is the modern system, and is best understood in our own country and in our own times.
...When those countries became commercial, the merchants found this prohibition [upon trading precious medals as capital], upon many occasions, extremely inconvenient. They could frequently buy more advantageously with gold and silver than with any other commodity the foreign goods which they wanted, either to import into their own, or to carry to some other foreign country. They remonstrated, therefore, against this prohibition as hurtful to trade.

They [the merchants] represented, first, that the exportation of gold and silver in order to purchase foreign goods, did not always diminish the quantity of those metals in the kingdom. That, on the contrary, it might frequently increase that quantity; because, if the consumption of foreign goods was not thereby increased in the country, those goods might be re-exported to foreign countries, and, being there sold for a large profit, might bring back much more treasure than was originally sent out to purchase them. Mr. Mun compares this operation of foreign trade to the seed-time and harvest of agriculture.

If we only behold, [says he] the actions of the husbandman in the seed-time, when he casteth away much good corn into the ground, we shall account him rather a madman than a husbandman. But when we consider his labours in the harvest, which is the end of his endeavours, we shall find the worth and plentiful increase of his action.
...Such as they were, however, those arguments convinced the people to whom they were addressed. They were addressed by merchants to parliaments and to the councils of princes, to nobles and to country gentlemen, by those who were supposed to understand trade to those who were conscious to themselves that they knew nothing about the matter. That foreign trade enriched the country, experience demonstrated to the nobles and country gentlemen as well as to the merchants; but how, or in what manner, none of them well knew. The merchants knew perfectly in what manner it enriched themselves. It was their business to know it. But to know in what manner it enriched the country was no part of their business.

It would be inaccurate to say that Marx just lifted Smith wholesale, but there is a certain parity that can be made. For instance, the Manifesto by Marx essentially explains the process similarly:

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The discovery of America, the rounding of the Cape, opened up fresh ground for the rising bourgeoisie. The East-Indian and Chinese markets, the colonisation of America, trade with the colonies, the increase in the means of exchange and in commodities generally, gave to commerce, to navigation, to industry, an impulse never before known, and thereby, to the revolutionary element in the tottering feudal society, a rapid development.

The feudal system of industry, in which industrial production was monopolised by closed guilds, now no longer sufficed for the growing wants of the new markets. The manufacturing system took its place. The guild-masters were pushed on one side by the manufacturing middle class; division of labour between the different corporate guilds vanished in the face of division of labour in each single workshop.

Meantime the markets kept ever growing, the demand ever rising. Even manufacturer no longer sufficed. Thereupon, steam and machinery revolutionised industrial production. The place of manufacture was taken by the giant, Modern Industry; the place of the industrial middle class by industrial millionaires, the leaders of the whole industrial armies, the modern bourgeois.

Modern industry has established the world market, for which the discovery of America paved the way. This market has given an immense development to commerce, to navigation, to communication by land. This development has, in its turn, reacted on the extension of industry; and in proportion as industry, commerce, navigation, railways extended, in the same proportion the bourgeoisie developed, increased its capital, and pushed into the background every class handed down from the Middle Ages.

We see, therefore, how the modern bourgeoisie is itself the product of a long course of development, of a series of revolutions in the modes of production and of exchange.

Marx was interested in this process, how this occurred. Feudalism came from somewhere, and it was probably a similar process, so it's applied further out.

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It seems to me that the position that Ancient Greece was simply a capitalist country is less historical than overtly political. If one can say that capitalism is the natural state of existence that everyone practiced and Marx perverted it, then you can score a propaganda point. However, nothing we know about ancient Athens concerns fiat systems, and Adam Smith himself notes a transition.

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So while I cannot say with certainty that there is no historian out there that claims that Ancient Athens was a capitalist country, it seems problematic.