The answer here is very simple:
You're already containing it in double quotes, so there's no need to escape it with \.
If you want to escape single quotes in a single quote string:
var string = 'this isn\'t a double quoted string';
var string = "this isn\"t a single quoted string";
// ^ ^ same types, hence we need to escape it with a backslash
or if you want to escape \', you can escape the bashslash to \\ and the quote to \' like so:
var string = 'this isn\\\'t a double quoted string';
// vvvv
// \ ' (the escaped characters)
However, if you contain the string with a different quote type, you don't need to escape:
var string = 'this isn"t a double quoted string';
var string = "this isn't a single quoted string";
// ^ ^ different types, hence we don't need escaping