Quakers do not have doctrine and as such there would be no such thing as a Quaker doctrine of the Trinity. Yet, many Quakers would have thoughts on the concept. Richard Rohr is a Fransiscan and not a Quaker but I would adhere to his sensibilities below. Glenn Morison (Winnipeg Monthly Meeting)
Niels Bohr, the Danish physicist who was a major contributor to quantum physics and nuclear fission, said the universe is “not only stranger than we think, but stranger than we can think.” Our supposed logic has to break down before we can comprehend the nature of the universe and the bare beginnings of the nature of God. I think the doctrine of the Trinity is saying the same thing. The “principle of three” breaks down all dualistic either-or thinking and sets us on a dynamic course of ongoing experience.
There are some things that can only be known experientially, and each generation must learn them for themselves. The “prayer of quiet” is a most simple and universal path. Of all the religious rituals and practices I know of, nothing will lead us to that place of nakedness and vulnerability more than regular experiences of solitude and silence, where our ego identity falls away, where our explanations don’t mean anything, where our superiority doesn’t matter and we have to sit there in our naked “who-ness.”
If God wants to get through to us, and the Trinitarian Flow wants to come alive in us, that’s when God has the best chance. God is not only stranger than we think, but stranger than the logical mind can think. Perhaps much of the weakness of the first two thousand years of reflection on the Trinity, and many of our doctrines and dogmas, is that we’ve tried to do it with a logical mind instead of with prayer. The belief in God as a Trinity is saying God is more an active verb than a stable noun. You know it in the flow of life itself.