To nurse (i.e. to feed at a breast) in German is stillen (both in the transitive and intransitive senses: Die Mutter stillte ihr Kind "The mother nursed her child"; Das Kind stillte an der Brust "The child nursed at (her mother's) breast", though the second form is more uncommon, I'd say)—apparently a derivative of still "still, quiet": by my feeling, a causative "to cause to be quiet". Which is what it often does, even if the child is not hungry but just needs some comfort or closeness :)
The cognate verb to still isn't used in English, though, is it? The only thing that comes to mind is technical jargon from Robert Jordan's Wheel of Time series; IIRC it's used for the process of removing a male channeler's powers, or something like that? Anyone familiar with the series who can provide details?