There is no official datepicker for bootstrap; as such, you should explicitly state which one you're using.
If you're using eternicode/bootstrap-datepicker, there's a startDate option. As discussed directly under the Options section in the README:
All options that take a "Date" can handle a Date object; a String formatted according to the given format; or a timedelta relative to today, eg '-1d', '+6m +1y', etc, where valid units are 'd' (day), 'w' (week), 'm' (month), and 'y' (year).
So you would do:
$('#datepicker').datepicker({
startDate: '+1d'
})