What is the use of Pattern.quote method?

The Pattern.quote method quotes part of a regex pattern to make regex interpret it as string literals.

Say you have some user input in your search program, and you want to regex for it. But this input may have unsafe characters so you can use

Pattern pattern = Pattern.compile(Pattern.quote(userInput));

This method does not quote a Pattern but, as you point out, wraps a String in regex quotes.

\Q means "start of literal text" (i.e. regex "open quote")
\E means "end of literal text" (i.e. regex "close quote")

Calling the Pattern.quote() method wraps the string in \Q...\E, which turns the text is into a regex literal. For example, Pattern.quote(".*") would match a dot and then an asterisk:

System.out.println("foo".matches(".*")); // true
System.out.println("foo".matches(Pattern.quote(".*"))); // false
System.out.println(".*".matches(Pattern.quote(".*"))); // true

The method's purpose is to not require the programmer to have to remember the special terms \Q and \E and to add a bit of readability to the code - regex is hard enough to read already. Compare:

someString.matches(Pattern.quote(someLiteral));
someString.matches("\\Q" + someLiteral + "\\E"));

Referring to the javadoc:

Returns a literal pattern String for the specified String.

This method produces a String that can be used to create a Pattern that would match the string s as if it were a literal pattern.

Metacharacters or escape sequences in the input sequence will be given no special meaning.