If you want to perform a server side function periodically and according to a timed interval, then you most likely want a client-side process. Judging from your question, however, it doesn't look like you need anything from the server-side at all. If this is just a value that increments according to a timer, why not just create a javascript function to do this?
If, on the other hand, you have a variable that somehow changes server-side and you need to pass it to the client, you would do something similar to the code below, except you would place an API call within the timer's handler instead of the incrementing value. The handler would call the API (which is basically just a function on the server-side that can be called by the client without a post-back), the API would perform the function and return a value, and the javascript would insert the value into the element on the web page. Granted, if you are new, there can be some complex steps involved when creating APIs, but that is how you would do it.
I'll include some links at the bottom of this answer to get you started on building the API if that's what you want to do.
Javascript
$(document).ready(function () {
var myAspTextbox = $('#myAspTextbox'); // variable for our ASP Textbox control.
var inc = parseInt(myAspTextbox.val() || 0); // variable for ou incrementing value. This will pull either (1) the value already in the textbox if one is there when the page is loaded, or (2) will set the value to zero.
var timeInterval = 2000 // set interval timer for two seconds
var incrementalInterval = setInterval(function () {
inc++;
myAspTextbox.val(inc);
}, 1500);
});
HTML
<html xmlns="http://www.w3.org/1999/xhtml">
<head runat="server">
<title>DEMO</title>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.3/jquery.min.js"></script>
<script type="text/javascript" src="freestyle.js"></script>
</head>
<body>
<form id="form1" runat="server">
<div>
<asp:TextBox ClientIDMode="Static" ID="myAspTextbox" runat="server" placeholder="Incremental Interval"></asp:TextBox>
</div>
</form>
</body>
</html>
If you want to create an API controller...
Whether you're using webforms or MVC, API controllers are an excellent tool for partial page rendering and dynamic content, and I always encourage budding developers to consider using them. The instructions for creating an API controller are a little vast to post here directly, but here are some excellent links that should get you started: