A root in the Qenya Lexicon of the 1910s given as ᴱ√QḶŘḶ [QḶÐḶ] “quiet” with derivatives like ᴱQ. qilde “quiet, rest, hush” and ᴱQ. qildi- “rest, stay quiet, be quiet” (QL/78); immediately below it had a variant form ᴱ√QILI² (QL/77). It had derivatives in the contemporaneous Gnomish Lexicon as well such as G. cwîl “quiet, peaceful, gentle” and G. cwildred “bat (animal)” (GL/28).
However, in a table of primitive forms elsewhere in the Gnomish Lexicon, primitive qḷd was deleted (GL/23) and in yet another section Tolkien gave a distinct base form gwil-² with derivatives like G. gwîl “cessation, quiescence, leaving off” and G. gwilthi “peace”, with a possible primitive form ᴱ√u̯il marked by Tolkien with a “?” (GL/45). Tolkien compared G. gwil-² to Q. qil-, so perhaps in this instance he imagined primitive initial gʷ > kʷ in Qenya, following the pattern whereby initial voiced stops were unvoiced.
Neo-Eldarin: I think it is worth positing a Neo-Eldarin root ᴺ√KWILID “quiet” to preserve these earlier Quenya words.