Antiochus, his representative at the Court of Arcadius, openly wrote in favor of the persecuted sect; and the encouragement received from this high quarter rapidly increased the number of professing Christians in the Persian territories.
"Quarter" has a meaning of "division of a town". For example you will sometimes hear "the Latin quarter". More generally "quarter" can mean "section of society" (though this use is dated and rare). You might say
There is concern from many quarters about the proposed business restructuring.
The Court of (the Eastern Roman Emperor) Arcadius was a very high ranking part of society. This court is the "High quarter".
The "persecuted sect" is the Christian community, and Antiochus wrote letters that were in favour of the Christians. He wrote these letters openly. That means he signed his name on the letters, instead of writing them secretly with no signature.