Scholars of religion usually consider Anabaptists and Quakers part of the "Radical Reformation", along with Unitarians, but the groups are not necessarily organically related.
Quakers actually came out of the English Puritan movement, a group called Spiritualists. The Spiritualists emerged around the English Civil War, which was a period where a variety of radical religious groups were springing up (the Diggers and the Levelers, who believed in land redistribution and egalitarianism). Early Spiritualists were agnostic on the religious questions of their day among Puritans (mostly around baptism and the order of worship) and became spiritual seekers, rejecting outward forms of religion, hence the name. Eventually, they gravitated around preachers like George Fox, but its not likely that Fox would have had much of a following if it wasn't for the wider Spiritualist movement within Puritanism.
Mennonites and Anabaptists have quite distinct origins, actually being more influenced by the early Swiss Reformation under Hyuldrich Zwingli (though Zwingli later went on to attempt to suppress their teachings, which were more radical than his own).
The "peace church" designation is more of an American or English term. The stance against warfare is simply an extension of pre-Constantinian Christian ethics, and is by no means uniform or without nuance. In reality, early groups like the Anabaptists in their confessions affirm the right of the state to have a monopoly on violence for just purposes- they just believe it isn't the calling of the Church to be part of it.
www.anabaptistwiki.org
Likewise, while Quakers have a widespread peace testimony, they also rely upon individual conscience, and there have been individual Quakers that have chosen to participate in war.
