There is no need to do it in Laravel at all...Laravel/PHP code runs server-side. The Laravel code just renders some HTML and JavaScript into the browser to get fullCalendar to work. Therefore you can do this entirely client-side using JavaScript, and disregard the Laravel part. The only thing you need to take note of from your PHP code is the ID you give to the calendar - replace the calendar
ID in my code below with the ID you've used.
HTML:
Date: <input type="date" id="dateField"/>
<button type="button" id="dateBtn">Go to date</button>
JavaScript:
$("#dateBtn").click(function(){
$("#calendar").fullCalendar("gotoDate", $("#dateField").val());
});
See http://jsfiddle.net/sbxpv25p/826/ for a working demo.