You can escape quotes/characters by prepending \
to it:
var string = 'my string with "double quotes" and \'single quotes\'';
var string = "my string with 'single quotes' and \"double quotes\"";
// ^ ^
Using a dynamic string:
var foo = "bar with 'quotes'";
var string = 'my string with "double quotes" and ' + foo.replace(/'/g, "\\'");
//my string with "double quotes" and bar with \'quotes\'