Both Quieter and More quiet are correct. Quieter is more common but more quiet is also used in modern English. Quieter is the comparative form of quiet.
Quieter often sounds more natural to English speakers and more quiet can sometimes sound strange.
The students are quieter than usual.
The students are more quiet than usual.
I would nearly always use quieter. It sounds more natural to my ear as a native English speaker from Ireland.
Sometimes when we are speaking, and our brains are trying to find the correct word, we say “I need something more…..quieter”.
In this example, we say more before we have fully formed the sentence in our mind and for that reason, we use more quieter. I notice this happens a lot with more difficult/harder.
Quieter as a comparative adjective
A comparative adjective compares two nouns.
When we form a comparative adjective, we use +er when the word is one syllable and more+adjective when it is two syllables or more.
Some adjectives like quiet and clever have two syllables but both forms are acceptable.
Quietest or the most quiet? Which is correct?
Quietest is a superlative adjective. A superlative adjective compares one noun to all of the other nouns in its category.
The quietest and the most quiet are both acceptable but “the quietest” is much more common.
The most quiet often sounds strange to the ear of a native English speaker.
Quieter or less Noisy?
There is an argument that the meaning of quiet suggests that there is no noise at all and therefore you can’t compare nouns as the result is binary(quiet or not quiet).
Most people ignore this and accept that something can be quiet and still make a little bit of noise.
This is the quietest gun on the market.
This is the least noisy gun on the market.
If you worked in marketing, you would definitely use the first example as it promotes the idea that the gun is quiet.
As quiet as
Another way to compare things in English is to use as+adjective+as+a+noun.
There is a very famous idiom “as quiet as a mouse”.
This construction can sound better than quieter or more quiet.
This laptop is not as quiet as this one.