Which version of the Gospels do the authors of the Philokalia refer to? Which version of the Gospels do the authors of the Philokalia (such as Gregory Palamas) refer to? Clearly this would be some Greek version. 

But I would like to know what would be the closest thing to that version, today. Both original greek versions and translation from those would be an acceptable answer. Thanks!
It is almost impossible that the writers of The Philokalia consulted exactly the same Gospel manuscripts.  It is a collection of writings from over 20 different authors spanning almost 1,000 years from different locations around the Mediterranean and Middle East.

For example: Isaiah the Solitary lived in Gaza in the 5th century; John Cassian lived in Gaul in the 4th and 5th centuries; John of Damascus in Syria in the 8th century; Gregory Palamas in Greece and Constantinople in the 14th century.

The 1904 Patriarchal Text mentioned in the comments was compiled from a number of disparate Greek texts and lectionaries that had been in use over the last two millennium.  It does not claim itself to be any kind of "original text".  In content it ends up being closest to the Majority Text.