Inspired by this, I translated the whole solution into JS. I renamed your "query" object to "output_state," which seems like a more accurate name.

    var output_state = {};
    function print(x) {
        console.log(x);
        if (typeof x === "number") {
            output_state = {last_num: x, time_since: 1};
        } else {
            output_state.time_since += 1;
        }
    }

    function fizzbuzz() {
        switch(output_state.last_num) {
            case undefined:
                print(1);
                break;
            default:
                var a = output_state.last_num;
                var b = output_state.time_since;
                var number = a + b;

                if (number % 5 !== 0) {
                    if (number % 3 === 0) {
                        print("Fizz");
                    } else {
                        print(number);
                    }
                } else {
                    if (number % 3 === 0) {
                        print("FizzBuzz");
                    } else {
                        print("Buzz");
                    }
                }
        }

        fizzbuzz();
        // Or, correctly: if (number < 100) { fizzbuzz(); }
    }