I am not sure on the reasons why the Turks moved into Anatolia, since this is an area of my timeline I have not researched properly yet, but I've seen some depictions of them migrating into Hungary instead.
As Salvador79 said, when we say "Turks" for Middle Ages, we're actually speaking of a large set of peoples, often called "Türk/Turkic/Turkish" because of their relations (linguistical or cultural) with Turks but sometimes as well unrelated people as Maygars.
In Europe, Turkic peoples were essentially represented by Avars, Old Bulgars, Pechenegs or Cumans/Kiptchak that if Turkic peoples, were quite distinct from islamized (and persianized) Turks of Anatolia.
The latter migrated there, but not directly. See Turkic mercenaries and slaves come from islamized turks peoples from Central Asia, essentially influenced by the Arabo-Persan civilisation, and began to form states by taking over their former masters' as Ghaznavids, or being pushed by other Turks forming their own states (Kara-Khanids, Oghuz) such as it happened for Seljuks.
Eventually, it's less Seljuks migating from Central Asia to Anatolia, than taking over the whole eastern Arabo-Islamic world. Not that is was an unifed states : dominated Arab states cohabited with state under Turkic domination with an overall Seljuk suzerainty.
Why Seljuks in Anatolia precisely? Well Byzantine Empire was still one of the major opponents of the Arabo-Islamic world, and its avance on Armenia and Middle-East was enough trouble for that Seljuks that while they knew a bit the region (critically on Caucasus), they attacked an empire that had already problems with Normans and Pechenegs, and managed to swallow all Anatolia by the late XIth century (that Byzantines would recover partially eventually).
Would it really be a possibility, considering the large Hungarian population already there, to from them to displace them and become the dominant ethnicity in the region?
I don't think : it would have meant for Hungarians to simply trample over Byzantines to cross the sea and Balkans and would have they done that, they wouldn't have left for a less wealthy and interesting Anatolia (let's be straight, for Hungarians to take on Byzzies would be insanely hard).
As far as I know : there wasn't an Hungarian population in Anatolia (if it's what you mean by "already there")
If you meant for Turks to take over Pannonian Plain...Well they did with Avars and partially so with Old Bulgars. But eventually, you had to have some form of stabilisation, and Hungarians did that pretty well. If they wouldn't have...My bet would be on Pechenegs, but not as successfully, would it be only for Germans and HRE being clearly the strong man of Central Europe.
Is it possible they could move to the Caucasus, or maybe Persia? Or could they even just stay in Central Asia?
They did so IOTL, it's just that Persia or Mesopotamia were definitely too big of cultures to be turkified as was (and in the long run) the anatolian melting pot.
Would it remain Greek or would it be picked off by one local power or another?
Thing is, Byzantine Anatolia wasn't wholly greek. You had Armenians, Syrians, Assyrians, a whole set Caucasians, already some Turks, etc.
While Byzantine identity managed (more or less, especially on on eastern Anatolia) to gather all of that, you had some distinct cultures. And I'm not even speaking of religious differences.