I heard the following dialog in a British English movie:
Words go round quick. All the little tongues go clack, clack, clack.
What does this mean, and why was the word "quick" used instead of "quickly"?
I see quickly as the correct adverb; I would correct my children if they said:
He's running quick.
However quick is widely used as an adverb, and in some phrases seems to work better:
a get rich quick scheme
Do it, and do it real quick
Some dictionaries do include that quick may be used as an adverb.
President George W Bush
We want to get this bridge rebuilt as quick as possible.
At to the meaning of the passage. The idea is of information (and suspect in particular malicious gossip) being transmitted very quickly from person to person. The individual tongues working very quickly like components of a machine.
This comes to mind:
A lie can travel halfway around the world before the truth can get its boots on
quick can be both adjective and adverb. In spoken language it may be used instead of quickly as an adverb mostly in exclamations or comparatives.
Come quick, Larry's on TV
Quick! there is a mouse
‘Quick’ (without -ly) is an adverb, as well as an adjective. An adverb does not have to end in -ly and often such adverbs that do not are called ‘flat adverbs’ (see Flat adverbs are flat-out useful).
‘Quick’ as an adverb has been used since 1300, per the Oxford English Dictionary (OED), which includes usages of the flat adverb ‘quick’ from Shakespeare, Milton, Locke, Chesterfield, Dickens, Tennyson, and Sandburg.
Do it quick, Come quick, and your Words go round quick are natural uses of quick as an adverb.
However, the OED also says that the use of quick as an adverb is
usually considered less formal than quickly, and found chiefly in informal or colloquial contexts [my emphases].
One might also add dialectal to that sentence, as its acceptance and/or usage in formal contexts may depend on the dialect of the speaker.