The #getOption method only returns options that you set - there is no way to get the specific list of values from the control (at least, not according to the documentation). You can, however, group the DataTable on the filtered column to get the total number of unique values, and use that instead:
google.visualization.events.addListener(personPicker, 'ready', function() {
// get column index, can also use filterColumnLabel, if you set that option instead
// if you use filterColumnLabel, you will need to translate it back into an index for the group
var columnIndex = personPicker.getOption('filterColumnIndex');
var group = google.visualization.data.Group(data, [columnIndex]);
var numpersons = group.getNumberOfRows();
alert('numpersons=' + numpersons);
if (numpersons > 1) {
$('div#personfilter').css('display', 'inline-block');
$('div#personfilter').css('visibility', 'visible');
} else {
$('div#personfilter').css('display', 'none');
$('div#personfilter').css('visibility', 'hidden');
}
});