Hi  asp777,
Refer below sample.
HTML
<script src="scripts/jquery-1.3.2.min.js" type="text/javascript"></script>
<script type="text/javascript">
    function Write() {
        $.ajax({
            type: "POST",
            url: "Default.aspx/Write",
            data: '{name: "' + $("#<%=txtUserName.ClientID%>")[0].value + '" }',
            contentType: "application/json; charset=utf-8",
            dataType: "json"
        });
    }
</script>
<div>
    Your Name :
    <asp:TextBox ID="txtUserName" runat="server"></asp:TextBox>
    <input id="btnGetTime" type="button" value="Write" onclick="Write()" />
    <asp:Button Text="Read" runat="server" OnClick="Read" />
    <asp:Button Text="Refresh" runat="server" OnClick="Refresh" />
</div>
Namespaces
C#
using System.Web.Services;
VB.Net
Imports System.Web.Services
Code
C#
protected void Page_Load(object sender, EventArgs e)
{
    if (!this.IsPostBack)
    {
        if (Request.Cookies["Name"] != null)
        {
            HttpCookie nameCookie = Request.Cookies["Name"];
            nameCookie.Expires = DateTime.Now.AddDays(-1);
            Response.Cookies.Add(nameCookie);
        }
    }
}
protected void Refresh(object sender, EventArgs e)
{
    Response.Redirect(Request.Url.AbsoluteUri);
}
[WebMethod]
public static void Write(string name)
{
    HttpCookie nameCookie = new HttpCookie("Name");
    nameCookie.Values["Name"] = name;
    HttpContext.Current.Response.Cookies.Add(nameCookie);
}
protected void Read(object sender, EventArgs e)
{
    HttpCookie nameCookie = HttpContext.Current.Request.Cookies["Name"];
    string name = nameCookie != null ? nameCookie.Value.Split('=')[1] : "";
    ClientScript.RegisterStartupScript(this.GetType(), "alert", "alert('" + name + "');", true);
}
VB.Net
Protected Sub Page_Load(ByVal sender As Object, ByVal e As EventArgs) Handles Me.Load
    If Not Me.IsPostBack Then
        If Request.Cookies("Name") IsNot Nothing Then
            Dim nameCookie As HttpCookie = Request.Cookies("Name")
            nameCookie.Expires = DateTime.Now.AddDays(-1)
            Response.Cookies.Add(nameCookie)
        End If
    End If
End Sub
Protected Sub Refresh(ByVal sender As Object, ByVal e As EventArgs)
    Response.Redirect(Request.Url.AbsoluteUri)
End Sub
<WebMethod()>
Public Shared Sub Write(ByVal name As String)
    Dim nameCookie As HttpCookie = New HttpCookie("Name")
    nameCookie.Values("Name") = name
    HttpContext.Current.Response.Cookies.Add(nameCookie)
End Sub
Protected Sub Read(ByVal sender As Object, ByVal e As EventArgs)
    Dim nameCookie As HttpCookie = HttpContext.Current.Request.Cookies("Name")
    Dim name As String = If(nameCookie IsNot Nothing, nameCookie.Value.Split("="c)(1), "")
    ClientScript.RegisterStartupScript(Me.GetType(), "alert", "alert('" & name & "');", True)
End Sub
Screenshot
