Following this article: Performance Boost from pg-promise library, and its suggested approach:
// Concatenates an array of objects or arrays of values, according to the template,
// to use with insert queries. Can be used either as a class type or as a function.
//
// template = formatting template string
// data = array of either objects or arrays of values
function Inserts(template, data) {
    if (!(this instanceof Inserts)) {
        return new Inserts(template, data);
    }
    this.rawType = true;
    this.toPostgres = function () {
        return data.map(d=>'(' + pgp.as.format(template, d) + ')').join(',');
    };
}
An example of using it, exactly as in your case:
var users = [['John', 23], ['Mike', 30], ['David', 18]];
db.none('INSERT INTO Users(name, age) VALUES $1', Inserts('$1, $2', users))
    .then(data=> {
        // OK, all records have been inserted
    })
    .catch(error=> {
        // Error, no records inserted
    });
And it will work with an array of objects as well:
var users = [{name: 'John', age: 23}, {name: 'Mike', age: 30}, {name: 'David', age: 18}];
  
db.none('INSERT INTO Users(name, age) VALUES $1', Inserts('${name}, ${age}', users))
    .then(data=> {
        // OK, all records have been inserted
    })
    .catch(error=> {
        // Error, no records inserted
    });
UPDATE-1
For a high-performance approach via a single INSERT query see Multi-row insert with pg-promise.
UPDATE-2
The information here is quite old now, see the latest syntax for Custom Type Formatting. What used to be _rawDBType is now rawType, and formatDBType was renamed into toPostgres.