It occurred in his "A Vindication of the Moravians, against the Aspersions of their Enemies,” The Independent Reflector, January 4, 1753.
He refers to "Popes" and "Persecutors" as being from "Rome, England, Holland or Geneva." The first two refer to Roman Catholic and Anglican Churches. The latter two refer certainly to Calvinistic Presbyterians and perhaps (Dutch) Arminians.
Livingston was already at this time -- when he defended "primitive Christianity" against "orthodoxy" -- formally/nominally affiliated with the Presbyterians. He joked about this when he wrote his satirical attack on the Anglicans' 39 Articles of faith and prefaced it with the aside:
It is well known that some have represented me as an Atheist, others as a Deist, and a third sort as a Presbyterian. My creed will show that none have exactly hit it. For all which reasons, I shall cheerfully lay before you the articles of my faith. * * *Livingston's "articles" were written around the same time as his defense of the Morvaians, 1753.
I don't see any evidence he changed his position over time. I have shown Livingston continued to dislike the Athanasian Creed until the end of his life. I have also shown that in 1778, while Governor of New Jersey, Livingston is describing himself as "more than half a Quaker" while decrying Protestant Popery that persecutes.
I briefly encountered some sources, while researching this post, that spoke of a Presbyterian-Quaker "meeting of the minds" theology that occurred during this time. I'm still trying to wrap my mind around that one. But then again, the Presbyterians gave birth to both the Socinian Joseph Priestley and the Calvinist turned Arminian turned Universalist Benjamin Rush.