For this, you need to create one more button 'Today Button' and create one function for the today button and write your logic on this.
todayDate() {
this.configMonth.startDate = DayPilot.Date.today().firstDayOfMonth();
this.configMonth.days = this.configMonth.startDate.daysInMonth();
let dt = new DayPilot.Date(this.configMonth.startDate);
this.dateString = dt.toString("MMMM yyyy")
}