The answer to your question is both "yes" and "no."

There is no evidence to suggest that any ancient Greek person during the Classical Period (lasted c. 490 – c. 323 BCE) ever had any concept of "germs." The germ theory of disease would not be developed until at least centuries later and it would not become mainstream medical science until the nineteenth century. Instead, the ancient Greek Hippokratic medical tradition held that most diseases were caused by μίασμα (míasma), meaning "pollution," which the Hippokratic doctors conceived as noxious vapors in the air.

Nonetheless, at least some ancient Greek people during the Classical Period were certainly well aware of the fact that some diseases are contagious and capable of spreading from one person to another person in close proximity. Probably the most famous piece of evidence for this comes from the Athenian historian Thoukydides (lived c. 460 – c. 400 BCE), who gives a very detailed and vivid description of the spread of the plague that struck Athens in 430 BCE in his Histories of the Peloponnesian War 2.47–54. In this description, he very clearly demonstrates at several points that he is aware that the plague could be transmitted from one person to another.

First, in his Histories of the Peloponnesian War 2.47.4, Thoukydides describes how the physicians were powerless to stop the plague and they died in the greatest numbers because they were the ones spending the most time with those who were infected. He writes, as translated by Richard Crawley:

"Neither were the physicians at first of any service, ignorant as they were of the proper way to treat it, but they died themselves the most thickly, as they visited the sick most often; nor did any human art succeed any better."

Later, in 2.50, he describes how the plague was apparently transmissible to scavenger birds that fed on the corpses of the deceased (again, in Crawley's translation):

"All the birds and beasts that prey upon human bodies, either abstained from touching them (though there were many lying unburied), or died after tasting them. In proof of this, it was noticed that birds of this kind actually disappeared; they were not about the bodies, or indeed to be seen at all. But of course the effects which I have mentioned could best be studied in a domestic animal like the dog."

Thoukydides's statements in Histories of the Peloponnesian War 2.51.4–6 suggest that he was not the only one who realized that the plague was transmissible from one person to another. He describes how most people were too terrified to tend to those who were infected because they knew that, if they came into too close of contact with the infected, they would become infected themselves. Even more fascinatingly, in the same passage, Thoukydides observes that those who had already caught the disease and recovered from it became immune to it. He writes (in Crawley's translation):

"By far the most terrible feature in the malady was the dejection which ensued when anyone felt himself sickening, for the despair into which they instantly fell took away their power of resistance, and left them a much easier prey to the disorder; besides which, there was the awful spectacle of men dying like sheep, through having caught the infection in nursing each other. This caused the greatest mortality."

"On the one hand, if they were afraid to visit each other, they perished from neglect; indeed many houses were emptied of their inmates for want of a nurse: on the other, if they ventured to do so, death was the consequence. This was especially the case with such as made any pretensions to goodness: honor made them unsparing of themselves in their attendance in their friends' houses, where even the members of the family were at last worn out by the moans of the dying, and succumbed to the force of the disaster."

"Yet it was with those who had recovered from the disease that the sick and the dying found most compassion. These knew what it was from experience, and had now no fear for themselves; for the same man was never attacked twice—never at least fatally. And such persons not only received the congratulations of others, but themselves also, in the elation of the moment, half entertained the vain hope that they were for the future safe from any disease whatsoever."

Finally, Thoukydides observes in his Histories of the Peloponnesian War 2.52.1–2 that the spread of the disease seemed to be exacerbated by the fact that all the Athenians from the countryside were packed inside the walls of the city at the time when the plague struck to keep them safe from the Spartans, with whom the Athenians were at war, and the plague was able to spread rapidly in the overcrowded city. He writes (as translated by Crawley):

"An aggravation of the existing calamity was the influx from the country into the city, and this was especially felt by the new arrivals. As there were no houses to receive them, they had to be lodged at the hot season of the year in stifling cabins, where the mortality raged without restraint. The bodies of dying men lay one upon another, and half-dead creatures reeled about the streets and gathered round all the fountains in their longing for water."

A. J. Holladay and J. C. F. Poole argue in their paper "Thucydides and the Plague of Athens" (published in The Classical Quarterly 29, no. 2 (1979): 282–300) that Thoukydides's observations of both contagion and acquired immunity are his own original contributions to the theory of disease.

I don't entirely buy this argument, especially since Thoukydides very much makes it sound like people in Athens at the time of the plague were widely aware that the plague was contagious and that people who had already recovered from it were immune. Nonetheless, Holladay and Poole are absolutely correct that Thoukydides does give a more accurate description of how diseases are transmitted than would be expected from a physician in the Hippokratic tradition.

Again, though, to be clear, Thoukydides has absolutely no concept of germs; all he really has is a vague sense that diseases can be transmitted from one person to another and that a person who has already had a particular disease acquires immunity to it.