The short answer is... no, you cannot.
But there are some alternatives, like 1) adding extra buttons to move 4-5 weeks ahead or back or 2) adding a button to select a date/week (for example with any datepicker widget)
Here you have the code (excerpt) for both options, assuming that you can use jquery's datepicker (or any other one) and you use v5 of FullCallendar:
var calendar = new FullCalendar.Calendar(calendarEl, {
[...]
headerToolbar: {
left: 'prev4w,prev,today,next,next4w opendatepicker',
center: 'title',
right: ''
},
customButtons: {
prev4w: {
//text: '-4w',
hint: '4 weeks back',
click: function(e) {
e.preventDefault();
calendar.gotoDate(moment(calendar.view.currentStart).subtract("4", "weeks").format("YYYY-MM-DD"));
},
},
next4w: {
//text: '+4w',
hint: '4 weeks ahead',
click: function(e) {
e.preventDefault();
calendar.gotoDate(moment(calendar.view.currentStart).add("4", "weeks").format("YYYY-MM-DD"));
},
},
opendatepicker: {
hint: 'Go to date...',
click: function(e) {
e.preventDefault();
if($("#datepicker").datepicker("widget").is(":visible")){
$("#datepicker").datepicker("hide");
}else{
$("#datepicker").datepicker("show");
}
},
}
},
datesSet: function(dateInfo) {
var inicio = moment(dateInfo.startStr);
[...]
$("#datepicker").val(inicio.format('DD/MM/YYYY'));
}
});
$().ready(function() {
var inicio = moment(calendar.view.currentStart).format("DD/MM/YYYY");
$(".fc-toolbar-chunk").first().append(
'<input type="hidden" id="datepicker" size="8"></input>'
);
$("#datepicker").datepicker({
[...]
dateFormat: "dd/mm/yy",
onSelect: function(formated, dates) {
var fecha = moment(formated,"DD/MM/YYYY",true).format('YYYY-MM-DD');
calendar.gotoDate(fecha);
},
[...]
});
$("#datepicker").datepicker( "setDate" , inicio );
})