replied on December 11, 2016
I can't promise this is the most efficient solution, but this should achieve what you are looking for.
This assumes that your dropdown has these choices and matching values:

It also assumes that your dropDown has a CSS Class of "dropDown", and your two date fields have CCS Classes of "startDate" and "endDate".
$(document).ready(function () {
//function is called when the selected option in the dropdown is called.
$('.dropDown').change( function () {
//get the selected value of the dropDown class which could be 1, 2, 3, or 4
var qtr = $('.dropDown :selected').val();
//calculate the first date of the quarter and populate into into the start date field.
var d1 = new Date();
var m1 = ((qtr * 3) - 3);
d1.setMonth(m1);
d1.setDate(1);
var d1x = d1.getDate() + '/' + d1.getMonth()+1 + '/' + d1.getFullYear();
$('.startDate input').val(d1x);
//calculate the last date of the quarter and populate into into the end date field.
var d2 = new Date();
var m2 = ((qtr * 3));
d2.setMonth(m2);
d2.setDate(0);
var d2x = d2.getDate() + '/' + d2.getMonth()+1 + '/' + d2.getFullYear();
$('.endDate input').val(d2x);
}) //end of $('.dropDown').change( function () {
}); //end of $(document).ready(function () {
I haven't completely tested this code as-is, because I have dates formatted as mm/dd/yyyy, but this code (as you requested it) has the dates formatted as dd/mm/yyyy. But when I had the values re-arranged it worked for me.