So while I can't speak to the Greek population specifically, I can talk a little but about conversion within Anatolia for the Armenian population (and having clarified via PM, this seems also relevant here). More specifically, as part of the Armenian genocide, while the killing of men, women, and children was common and widespread, Armenian identity was also assaulted through the forcible and coerced conversion of women and children, who were then incorporated into Turkish households (I would note this is specifically defined as genocide by the UN: "forcibly transferring children of the group to another group." Genocide is more than simply killing, it is about destruction of identity and group cohesion).

Exact estimates are not easy to come by, but the number of Armenians who converted to Islam during the genocide period is estimated to be in the range of 100,000 to 200,000 persons (for comparison estimates for those killed in the genocide range from 800,000 at minimum, to 1.5 million). The specifics of how varied by circumstance, and at least in some situations the Armenians were taken in by Turkish, Arab, or Kurdish friends who could offer them protection, although to do so they had to comply with the requirements of conversion and taking on a Turkish name. For the young boys, they also had to undergo circumcision. In others, there was at least the veneer of voluntary conversion, such as the Armenian community from Köyulhisar which petitioned to be allowed to convert en masse "of our own free will", but the choice of courrse must be understood in the context of the dire situation they faced, and Ottoman authories were very cautious in allowing such converstions which risked being less then sincere.

More generally though the distribution was hardly within such power of choice by the Armenian personages. They had no real say in where they might be placed, only the choice between uncertainty of deportation, and quite possibly a fatal destination, or conversion. And of course for children in orphanages, left behind by their parents being marched to death, they had no choice at all as the government redistributed them among Muslim households. Not, it should be said, to always be treated as loved and equal members of a family. For those old enough, they would end up as servants in their new homes. And for those unwanted... when closed down, the 700 or so children and their Armenian caretakers at the orphanage in Harput were reportedly killed by drowning in the Euphrates River.

On the "deportation" marches themselves though, Turks would literally peruse the lines of Armenians, and pick out women and children to 'absorb into their family'. Leslie Davis, the American consul in Harpoot, described the process, with wealthy men showing up even with doctors in tow to evaluate the health of their selectees, and favoring "the prettiest girls" who would now, presumably, become a junior wife in their household, or else simply a concubine.

Although not an entirely unknown part of the genocide, being partially documented by American consular officials and missionaries who witnessed it, as well as some extant official documentation on the redistribution as well, it certainly was well buried in Turkish memory, where the Genocide itself is suppressed and denied. Fethiye Cetin's "My Grandmother: An Armenian-Turkish Memoir" was thus published not without controversy, documenting the life of her grandmother, Seher, who, as a young girl, had been placed in a Muslim household and raised Turkish, having quite literally been taken from her mother's arms, and keeping quiet about her origins for most of her life, something her granddaughter was entirely unaware of:

I had grown up thinking of myself as a Turkish Muslim, not an Armenian. There had been nothing in the history books about the massacre of a people that had been erased from Turkey’s collective memory. Like my grandmother, many had buried their identity — and the horrors they had seen — deep inside of them.

Her's was a story with a somewhat happy ending though. Seher's parents were the lucky ones who survived, and managed to eventually emigrate to the United States. After discovering the story of her origins, Fethiye was able to get in touch with her Armenian relations then living in New York.

Ronald Grigor Suny, They Can Live in the Desert but Nowhere Else: A History of the Armenian Genocide

Ara Sarafian, "The Absorption of Armenian Women and Children into Muslim Households As a Structural Component of the Armenian Genocide" in In God's Name: Genocide and Religion in the Twentieth Century ed. Omer Bartov and Phyllis Mack

Dan Bilefsky, "A Family Tree Uprooted by a 60-Year-Old Secret" New York Times, Jan. 5, 2010