An implementation of the Vue Router `stringifyQuery` method which handles nested query parameters. It uses square brackets (`[]`) for query parameter arrays (regular and nested).

vue-router-stringify-query-nested-parameters.js

/*
Ex:
stringifyQuery({foo: "bar", baz: ['a','b'], fizz: {foo: [1,2,3], bar: {bees: "knees"}}});
-> ?foo=bar&baz[]=a&baz[]=b&fizz[foo][]=1&fizz[foo][]=2&fizz[foo][]=3&fizz[bar][bees]=knees

Square brackets will be URL encoded; I didn't do so in this example for readability.
*/

function stringifyQuery (obj, parentName) {
    const res = obj ? Object.keys(obj).map(key => {
        const val = obj[key];

        if (parentName) {
            key = parentName + '[' + key + ']';
        }

        if (val === undefined) {
            return '';
        }

        if (val === null) {
            return encodeURIComponent(key);
        }

        if (!Array.isArray(val) && typeof val === "object") {
            return stringifyQuery(val, key).slice(1);
        }

        if (Array.isArray(val)) {
            const result = [];

            val.forEach(val2 => {
                if (val2 === undefined) {
                    return;
                }
                if (val2 === null) {
                    result.push(encodeURIComponent(key + '[]'));
                } else {
                    result.push(encodeURIComponent(key + '[]') + '=' + encodeURIComponent(val2));
                }
            })

            return result.join('&');
        }

        return encodeURIComponent(key) + '=' + encodeURIComponent(val);
    }).filter(x => x.length > 0).join('&') : null;

    return res ? `?${res}` : '';
}

You also need a custom parseQuery function or this.$route.query will be { 'foo[]': ['bar', 'buzz'] } instead of { foo: ['bar', 'baz'] }