You can select a given day using .select() method:
var nav = DayPilot.Navigator("nav");
By default, it will also automatically bring the target month into focus and call onTimeRangeSelect/ed events.
If you want to synchronize the selection with other navigation methods (e.g. previous/next buttons) I recommend using the navigator as the primary date source. The next/previous buttons would call nav.select(). That would subsequently update the Scheduler using DayPilot.Navigator.onTimeRangeSelected event handler.