You cannot use ++ on something which is not a variable, this would be the closest you can get:
$('#counter').html(function(i, val) { return +val+1 });
jQuery's html() method can get and set the HTML value of an element. If passed a function it can update the HTML based upon the existing value. So in the context of your code:
$("#update").click(function() {
$('#counter').html(function(i, val) { return +val+1 });
}
DEMO: http://jsfiddle.net/marcuswhybrow/zRX2D/2/
When it comes to synchronising your counter on the page, with the counter value in your database, never trust the client! You send either an increment or decrement signal to you server side script, rather than a continuous value such as 10, or 23.
However you could send an AJAX request to the server when you change the HTML of your counter:
$("#update").click(function() {
$('#counter').html(function(i, val) {
$.ajax({
url: '/path/to/script/',
type: 'POST',
data: {increment: true},
success: function() { alert('Request has returned') }
});
return +val+1;
});
}