Get input value whenever changed

Try to use .keyup() , this will insert the link as soon as you type any number.

Try this here:

$('#quantity').keyup(function () {
  
  // Se não for número muda para nada 
  if(!$.isNumeric($(this).val())){
     $(this).val('');   
  }
  
  // Se for maior que máximo altera para o máximo
  if($(this).val() > $(this).attr('max')){
     $(this).val( $(this).attr('max') );   
  }
  
  // Se for menor que o minimo (mas diferente de nada, porque se apagar é nada) muda para o minimo
  if($(this).val() < $(this).attr('min') && $(this).val() != ''){
     $(this).val( $(this).attr('min') );   
  }
  
  // Se for diferente de nada ele muda o link para o obtido
  // Se for igual ao nada o link não será modificado
  if($(this).val() != ''){
     qty = $(this).val();
  }
  
  // Insere o link como TEXT, altere isto!
  // Apenas para visualização!
  $('demostracao').text('/busca/quantidade='+qty);
  
});

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script><inputid="quantity" name="quantity" min="1" max="3" type="text" value="1">
<br>
<demostracao></demostracao>

The basic function, without the extra fixes, is here:

$('#quantity').keyup(function () {
  qty = $(this).val();
  $('meu-botao').attr('href','/busca/quantidade='+qty);
}

The keyup will execute the function whenever the key is released, but this does not require the user to leave the field or click off the screen, as in the case of .on('change') mentioned.

Just add the value directly this way:

$('meu-botao').attr('href','/busca/quantidade='+$('#quantity').val())

This will always take the current value of the quantity element.

A question, unrelated to the focus of the question, why do not you just use "change"?

$('#quantity').on('change keyup paste click',function() {

It would look like this:

$('#quantity').on('change',function() {

If the intent is to get only when it modifies, the change event is sufficient, regardless of how the change comes in.

Now, continuing the focus of the question, if, with the suggested change $('#quantity').val() , the qty variable is no longer used, the change event is no longer meaningful. You could delete this snippet of code.

[edit]

Complete example:

$('#quantity').on('change',function() {
    $('meu-botao').attr('href','/busca/quantidade='+$('#quantity').val());
}

I think this should work:

<input id="quantity" name="quantity" min="1" max="3" type="number">
<div id="divRetorno"></div>

<script>
$('#quantity').on('change',function() {
    qty = $(this).val();
    $.ajax({
        type:'GET',
        url: "/busca/quantidade="+qty,
        success: function(result){
            console.log(result)
            $("#divRetorno").html(result);
        }
    });

});
</script>