To make it into a trite joke,
My co-worker Chris posed the question, it having been raised on a Unitarian Universalist discussion list. Here’s a quick stab at an answer using resources available to me at the office (which boils down to things on the web).
The current condition of the Religious Society of Friends is such that you can find Friends who believe most anything, and even Quaker bodies that endorse or tolerate quite a wide range. In addition, modern liberal Friends (the ones I’m most familiar with) just don’t do much theology. In general, though, I think most modern liberal Quakers do tend toward a unitarian understanding of God.
Historically, and for many if not most Quakers today, Friends have used the language of God/Father, Jesus/Christ/Son, and Spirit/Holy Spirit. (And some even have a developed theology of the offices of Christ!) The first Quaker systematic theologian, Robert Barclay, in his Apology for the True Christian Divinity, first published in 1678, doesn’t use the word “trinity” in this work, but on the other hand includes propositions like: “Concerning the Universal Redemption by Christ, and also the Saving and Spiritual Light wherewith every man is enlightened.” His proposition on immediate revelation begins:
“Seeing “no man knoweth the Father but the Son, and he to whom the Son revealeth him”; and seeing the “revelation of the Son is in and by the Spirit” (Matt. 11:27); therefore the testimony of the Spirit is that alone by which the true knowledge of God hath been, is, and can be only revealed; . . . .”
I can’t really do the research, but I recall that somewhere I’ve read early Quaker rebuttals of charges that they were unitarian, which say that “trinity” “trinitarian” and “three persons in one” are unbiblical language and therefore inappropriate (even if Quakers are, in fact, trinitarian).
Here’s a link to the Quaker Heritage Press version of Barclay’s Apology.