Is it correct to say “I will lie quiet beneath his touch”? Shouldn’t it be “I will lie quietly beneath his touch”?
As sumelic says in his answer, both "quiet" and "quietly" are grammatical here. "Quiet" is a complement of the verb "lie", while "quietly" is a manner adverb modifier of the verb phrase "lie". ("lie" is both a verb and the head of the single word verb phrase "lie".)
Though the sentence structures for the "quiet" and "quietly" versions differ, the meanings seem to be the same. The grammatical difference can be seen, however, in the greater mobility of the adverb: "He was quietly lying on the bed" but *"He was quiet lying on the bed" (ungrammatical with no comma intonation after "quiet").
Also, there can be only one adjective complement to "lie", so it is okay to have the adverb "quietly" modifying the verb phrase "lying asleep" in "He was quietly lying asleep", but "asleep" and "quiet" can't both be complements of "lie": *"He was lying quiet asleep."
"Lie quiet" and "lie quietly" are both grammatical, and I would say both are acceptable in standard English. There is probably some slight difference in meaning, but I don't think it's important in this context: either could be used here.
Here are a couple of examples of "lie quiet" from published sources, taken from Google Books:
The usually intolerable flies, seeking release through window openings covered with dust-ridden wire screen, fought momentarily to get through to open air, then dropped to the sill below to lie quiet.
Deep Enough: A Working Stiff in the Western Mine Camps, by Frank A. Crampton
Whenever it was moved it cried, but at all other times it was so patient that the sole desire of its life appeared to be to lie quiet and think.
Bleak House, by Charles Dickens
The question of why "lie quiet" is acceptable is interesting. I don't know how to explain it. It is certainly not true that "lie" can never be followed by an adjective. A Comprehensive Grammar of The English Language (Quirk, Greenbaum, Leech & Svartik) apparently includes the observation that lie, while much more restricted in scope than other copular verbs, can be followed by the adjective "flat" (Grammar Glossary references: copular verbs).
Quiet should be used in this scenarios, as Quietly is an adverb. Quite is used when u want add some expression to the action.
I was quiet in the meeting
The above one is a classic example where the Adjective "Quiet" helps to specify the behavior in the meeting.
Adverb - Quietly is used an action.
In your example "Quiet" is more apt as you are talking about the action that you are doing at that time.