Hi Warren,
It is not possible to check with your code. I have created small example, so that you can understand it.
Check this example. Now please take its reference and correct your code.
Database
I have made use of the following table Customers with the schema as follows.

I have already inserted few records in the table.

You can download the database table SQL by clicking the download link below.
Download SQL file
HTML
<asp:GridView ID="gvCustomer" runat="server" AutoGenerateColumns="False" DataKeyNames="CustomerId"
CssClass="table table-striped table-bordered table-hover">
<Columns>
<asp:BoundField DataField="CustomerId" HeaderText="ID" />
<asp:TemplateField HeaderText="Name">
<ItemTemplate>
<asp:TextBox runat="server" ID="txtName" Text='<%# Eval("Name") %>' CssClass="form-control Update" />
</ItemTemplate>
</asp:TemplateField>
<asp:TemplateField HeaderText="Country">
<ItemTemplate>
<asp:TextBox runat="server" ID="txtCountry" Text='<%# Eval("Country") %>' CssClass="form-control Update" />
</ItemTemplate>
</asp:TemplateField>
</Columns>
</asp:GridView>
<link rel="stylesheet" href="http://maxcdn.bootstrapcdn.com/bootstrap/3.3.2/css/bootstrap.min.css" />
<script type="text/javascript" src="https://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
<script type="text/javascript">
$(function () {
$('.Update').on('change', function () {
var id = $(this).closest('tr').find('td').eq(0).html();
var name = $(this).closest('tr').find('[id*=txtName]').val();
var country = $(this).closest('tr').find('[id*=txtCountry]').val();
$.ajax({
type: "POST",
url: "Default.aspx/Update",
data: "{id:" + id + ",name:'" + name + "',country:'" + country + "'}",
contentType: "application/json; charset=utf-8",
dataType: "json",
success: function (r) {
alert("Record updated successfully.");
},
error: function (r) {
alert(r.responsetext);
}
});
});
});
</script>
Namespaces
C#
using System.Configuration;
using System.Data;
using System.Data.SqlClient;
using System.Web.Services;
VB.Net
Imports System.Configuration
Imports System.Data
Imports System.Data.SqlClient
Imports System.Web.Services
Code
C#
protected void Page_Load(object sender, EventArgs e)
{
if (!IsPostBack)
{
gvCustomer.DataSource = GetData();
gvCustomer.DataBind();
}
}
private DataTable GetData()
{
string conString = ConfigurationManager.ConnectionStrings["constr"].ConnectionString;
string query = "SELECT CustomerId,Name,Country FROM Customers";
SqlCommand cmd = new SqlCommand(query);
using (SqlConnection con = new SqlConnection(conString))
{
using (SqlDataAdapter sda = new SqlDataAdapter())
{
cmd.Connection = con;
sda.SelectCommand = cmd;
using (DataTable dt = new DataTable())
{
sda.Fill(dt);
return dt;
}
}
}
}
[WebMethod]
public static void Update(int id, string name, string country)
{
string conString = ConfigurationManager.ConnectionStrings["constr"].ConnectionString;
string query = "UPDATE Customers SET Name = @Name,Country = @Country WHERE CustomerId = @Id";
using (SqlConnection con = new SqlConnection(conString))
{
SqlCommand cmd = new SqlCommand(query, con);
cmd.CommandType = CommandType.Text;
cmd.Parameters.AddWithValue("@Name", name);
cmd.Parameters.AddWithValue("@Country", country);
cmd.Parameters.AddWithValue("@Id", id);
con.Open();
cmd.ExecuteNonQuery();
con.Close();
}
}
VB.Net
Protected Sub Page_Load(ByVal sender As Object, ByVal e As EventArgs) Handles Me.Load
If Not IsPostBack Then
gvCustomer.DataSource = GetData()
gvCustomer.DataBind()
End If
End Sub
Private Function GetData() As DataTable
Dim conString As String = ConfigurationManager.ConnectionStrings("constr").ConnectionString
Dim query As String = "SELECT CustomerId,Name,Country FROM Customers"
Dim cmd As SqlCommand = New SqlCommand(query)
Using con As SqlConnection = New SqlConnection(conString)
Using sda As SqlDataAdapter = New SqlDataAdapter()
cmd.Connection = con
sda.SelectCommand = cmd
Using dt As DataTable = New DataTable()
sda.Fill(dt)
Return dt
End Using
End Using
End Using
End Function
<WebMethod()>
Public Shared Sub Update(ByVal id As Integer, ByVal name As String, ByVal country As String)
Dim conString As String = ConfigurationManager.ConnectionStrings("constr").ConnectionString
Dim query As String = "UPDATE Customers SET Name = @Name,Country = @Country WHERE CustomerId = @Id"
Using con As SqlConnection = New SqlConnection(conString)
Dim cmd As SqlCommand = New SqlCommand(query, con)
cmd.CommandType = CommandType.Text
cmd.Parameters.AddWithValue("@Name", name)
cmd.Parameters.AddWithValue("@Country", country)
cmd.Parameters.AddWithValue("@Id", id)
con.Open()
cmd.ExecuteNonQuery()
con.Close()
End Using
End Sub
Screenshots
The Form

Record After Update

Note: I have used inline query in the code for update. You need to change it with Stored Procedure.