Will Greece close its borders in the fall?

No one knows what any country will do in the immediate future. If I were you, I would keep abreast of any official statements from the Greek government (use sources like Reuters, AP, Bloomberg, or simply Google to find them). Oficial government statements and press releases will give the best signals of where things are heading (they are a better source than pure guesswork or travel forums). I don't think travel agents will have much more insights than anyone else (who is paying close attention) since everything is decided by foreign sovereign governments and can change on a dime....but they presumably follow the government guidance more than the common layperson. Also, ourworldindata publishes stats on covid cases and vaccinations rates - those also will give some clue, although they've become more and more disassociated with political and economic choices.

If you're thinking of booking a trip to Greece in the fall and want to avoid any covid-related issues with cancellations, book only fully-refundable flights, hotels, tours, etc. so if you're not permitted to go you at least won't have lost your money. Also try to find some travel insurance that will contribute to your protection against unforeseen issues.

And your travel agent has a crystal ball and graduated from "Hogwarts"?

Honestly I don't think anyone knows what is going to happen in 2 weeks much less "October"

My airfare to Europe is not refundable but Delta will give me credit to use until the end of 2022 and my hotels can be cancelled.

No one knows.

Those of us based in Europe on this forum are generally more cautious about Covid than the Americans.

It’s too early to cancel.

We are vaccinated Americans and have been traveling in Greece for a month, plan to stay another two months. Everything is good so far. Low numbers of tourists, although we are avoiding the areas that most tourists go to. Right now Covid is spiking in the party islands of Ios and Mykonos. Most other islands are fine.

We do our own bookings usually just a week or two in advance. I doubt if Greece will close to visitors in October but anything is possible. Best to avoid non refundable reservations.

I am in the same boat. Have a trip booked for mid-September. It is too early to say anything at this point.
So my strategy is to book only refundable hotels on booking.com, changeable airfare so I can postpone the trip and having trip interruption/cancellation insurance from my credit card (Chase Reserve).
In case they closed or Covid go out of control there I will start canceling.
But I am still optimistic: Greece vaccination rate already about 40%, will be probably by more than 50% by September, hospitalization and death are not picking up like before so I don't see a reason for tourism dependent country to shut down. By comparison, In New York State, 55% of the population is fully vaccinated.
They may shut it down for unvaccinated tourist, that possible, like they shut down indoor venues for unvaccinated citizens.
Surge in Covid cases in Greece not related to tourism: Minister

We certainly do live in uncertain times! When you book a trip, I would get traveler's insurance. I would also make sure that everything you book is refundable, just in case.

I would make sure you are vaccinated.

Those of us based in Europe on this forum are generally more cautious
about Covid than the Americans.

I just got back, and if you mean cautious in planning, I wouldn't know. But if you mean cautious in behavior, that wasn't my experience in the few countries I visited. Greece included.