jquery optimize ajax request? php and js

This is a basic debouncing problem. Change <input id="qty"> to <input class="qty"> and:

$('.button').click(function(e) {
    var btn = $(this), qty = btn.parent().find('.qty');

    clearTimeout(btn.data('timeout'));

    qty.val(qty.val()+1);

    btn.data('timeout', setTimeout(doPost(qty), 1000));
});

function doPost(el) {
     return function() {
         $.post('/ajax.php', {qty:el.val()});
     }
}

You will need to save the the clicks in a variable and wait a short amount of time after each click before making the final post.
Here is an implantation example, each time the user clicks he has another 100 milliseconds to click again, which is then added to the total clicks. only the last click causes the post event to occur, after which the click is reset back to 0.
During the post the button is disabled to prevent further clicking.

$(document).ready(function(){
    var clicks = 0;
    var timeoutId = 0;
    // putting the post in a separate function prevents unnecessary closures 
    var doPost = function(){
        $(".button").attr("disabled", "disabled");
        $.post('/ajax.php', "qty="+clicks, function(){
            $(this).parent().find(input[type='text']).val("1");
            clicks = 0;
            $(".button").removeAttr("disabled");
        });
    };
    $(".button").click(function(){
        clearTimeout(timeoutId);
        timeoutId = setTimeout(doPost, 100);
        clicks++;
    });
});

You can setTimeout for some small time (e.g. 1 sec) in .button#onclick handler that will call for $.post() (instead of posting values directly). Timeout handler must be saved into variable globally to all .button#onclick executions. Additionally every .button#onclick execution must check if previos timeout is still active. If it is, kill old timeout and setup new one. If not, just setup new.

So every button click will kill old timeout and set new timeout. Only the last-click setTimeout code body will be executed.

What do you mean by "fast"? Probably you want to use a timer function like Underscore's debounce (if you don't want to include whole underscore, you can cut it out and use it standalone):

<div class='qty'>
   <input type='text' id="qty_num" value='1'>
   <a class='button' id="qty_add" href='#'>add To Basket</a>
   <output id="qty_out">0</output><label for="qty_out"> items in basket</label>
</div>

$(document).ready(function() {
    var count = 0;
    function send() {
        $.post('/ajax.php', {qty: count}, function(result){
            $("#qty_out").val(result);
        });
        count = 0; // reset
    }
    var sendAfterSecond = _.debounce(send, 1000); // here
    $("#qty_add").click(function(e) {
         var num = parseInt($("#qty_num").val());
         count += num;
         sendAfterSecond();
    });
});

$(document).ready(function(){

    $(".button").click(function(){

    // disable the button
    $(this).attr("disabled", "disabled"); // if its a submit button

    var qty=$(this).parent().find(input[type='text']).val();

    // request some thing back in json {"success":1} or 0 if failed to add to cart.
    $.post('/ajax.php', "qty="+qty, function(m){ 
            if(m.success==1)
            {
                // enable the button
                $(".button").removeAttr("disabled"); // if its a submit button
            }

            $(this).parent().find(input[type='text']).val("1");
        },"json");

    });

});