Yes, Greeks enslaved other Greeks gladly and often.

The ancient world at large relied heavily on slavery and pretty much never questioned the institution. The Greek states were tied into a huge interregional slave trading network, and many of their slaves were not Greek; already in Homer we find an enslaved Sicilian woman sold to the Greeks by the Phoenicians, and the stereotypical enslaved person in Greek comedy came from Thrace or Phrygia. But this did not mean that the Greeks had any objections to enslaving other Greeks. They did not get their enslaved labourers from abroad out of principle, but simply bought what was available on the market.

In the Archaic period, growing income inequality seems to have driven many poor people within Greek states into debt bondage - that is to say, they were forcibly enslaved as they defaulted on their debts. The rich lenders who benefited from this system either used these enslaved debtors as land workers on their estates or sold them on the market. At Athens, debt bondage put such pressure on the poor that it nearly caused a civil war. This prompted the reforms of Solon (594 BC), abolishing debt bondage and (allegedly) bringing all Athenians sold abroad back to the city. Henceforth it was illegal for one Athenian to enslave another. At Sparta, meanwhile, debt bondage was formalised along with various other kinds of unfree status into the system of helotage, in which enslaved families would serve their rich overlords in perpetuity. It is only later that the Spartans started telling stories about how they were immigrants ruling over a subjected local population; nothing in the Archaic sources suggests this was really the case, and their state was built on the enslavement of Greeks by Greeks.

As Greek communities developed into more institutionalised states, many of them seem to have banned the practice of enslaving fellow citizens, like Solon had done at Athens. This was not out of a general solidarity with "the Greeks", but out of a more narrowly but legally defined concept of the in-group. It remained entirely legal, widely practiced, encouraged and admirable to enslave Greeks who were not fellow citizens.

In some cases this would be done by the legal system. Some crimes carried the punishment of enslavement. Especially immigrants (who were not citizens) were often enslaved if they were found guilty of certain crimes; at Athens, for example, the punishment for a foreigner marrying an Athenian citizen was enslavement. Citizens themselves could be enslaved too, as long as they were sold abroad, that is, to other Greeks or non-Greeks (so that no fellow citizen would commit the crime of enslaving them).

But by far the most common way for Greeks to enslave other Greeks was war. Neighbouring states would not just steal cattle in border raids, but also snatch human beings to sell into slavery. Prisoners of war and captives from taken cities were commonly enslaved; this was one of the main ways for states and individual commanders to profit from war. The Greeks seem to have held to a tacit rule that, while it was OK to massacre any non-Greek, the citizen women and children of Greek cities ought to be spared - but they were not spared the fate of enslavement. There are plenty of examples:

[The Spartans] took them out and slew them all without exception. The number of Plataians thus massacred was not less than two hundred, with twenty-five Athenians who had shared in the siege. The women were taken as slaves.

-- Thuc. 3.68.1-2

The Melians surrendered at discretion to the Athenians, who put to death all the grown men whom they took, and sold the women and children for slaves, and subsequently sent out five hundred colonists and inhabited the place themselves.

-- Thuc. 5.116.3-4

Xenophon described the terror of the Athenians at the thought of their city falling to the Spartans, and from their fear it becomes clear how frequently they had enslaved other Greeks in the short history of their Empire. It also shows that they expected no better from their enemies:

During that night no one slept, all mourning, not for the lost alone, but far more for their own selves, thinking that they would suffer such treatment as they had visited upon the Melians, colonists of the Spartans, after reducing them by siege, and upon the Histiaians and Skionaians and Toronaians and Aeginetans and many other Greek peoples.

-- Xenophon, Hellenika 2.2.3

Of course, there were occasional complaints about these atrocities. With the slow emergence of the notion of panhellenism - the idea that the Greeks should be united against the common enemy, the Persians - some people argued that it was wrong for any Greek to enslave another. But such claims are hollow. Consider the Spartan commander Kallikratidas, when he captured Methymna on Lesbos, an ally of Athens:

All the property which it contained the soldiers seized as booty, but all the captives Kallikratidas assembled in the marketplace; and when his allies urged him to sell into slavery the Methymnaians as well as the Athenians, he said that while he was commander no Greek should be enslaved if he could help it.

-- Xen. Hell. 1.6.14

This sounds admirable at first glance, but the request of the Spartan allies already makes clear what is actually at stake: not the question whether any Greeks should be enslaved, but whether the local population should also be enslaved along with the Athenian garrison. Kallikratidas' principle is therefore pretty limited:

Accordingly on the next day he let the Methymnaians go free, but sold the members of the Athenian garrison and such of the captives as were slaves.

-- Xen. Hell. 1.6.15

This is what went for admirable restraint among the Classical Greeks. Most other commanders would not hesitate to sell captives into slavery regardless of their origins, and it seems likely that enslaved Greeks would find willing buyers in the Greek world. After all, it was easier to order them around, since they already spoke the language.