As someone who was (loosely -- didn't go to meeting much, but was taught the philosophy and allowed to make raised in the Liberal Quaker tradition of Pennsylvania (loosely -- didn't go to meeting much, but was taught the philosophy and allowed to make what I would of it), I can say these things:
It is true that the early Quakers (or to use the official name: Friends) didn't use music or even a preacher in their meetings. Friends believe in a direct communication between God and the worshipper, and songs, stained glass, pulpits, etc. were all seen as a distraction from the act of truly listening to the Spirit Within.
However, it is also true that Friends believe that the Spirit of God speaks directly to the heart, and when the Spirit moves you, it is your obligation to act on it. So if a person is moved to sing, they sing.
And the lines:
"It finds an echo in my heart,
How can I keep from singing?"
is most definitely in keeping with the Friends belief.
So however unlikely it is to have a Quaker hymn, this may be a likelier candidate than others...