Hi  democloud,
Refer below sample.
HTML
<asp:DropDownList runat="server" ID="DropDownList1" AutoPostBack="true" OnSelectedIndexChanged="Change">
    <asp:ListItem>Select</asp:ListItem>
    <asp:ListItem>USA</asp:ListItem>
    <asp:ListItem>Austria</asp:ListItem>
    <asp:ListItem>Brazil</asp:ListItem>
</asp:DropDownList>
<br />
<br />
<asp:GridView ID="gvCustomers" runat="server" AutoGenerateColumns="false" AllowPaging="true"
    PageSize="5" OnPageIndexChanging="gvCustomers_onpageindexchanging">
    <Columns>
        <asp:BoundField DataField="CustomerId" HeaderText="CustomerId" />
        <asp:BoundField DataField="CompanyName" HeaderText="Name" />
        <asp:BoundField DataField="Country" HeaderText="Country" />
    </Columns>
</asp:GridView>
Namespaces
C#
using System.Data;
using System.Data.SqlClient;
using System.Configuration;
VB.Net
Imports System.Data.SqlClient
Imports System.Data
Code
C#
protected void Page_Load(object sender, EventArgs e)
{
    if (!this.IsPostBack)
    {
        string constr = ConfigurationManager.ConnectionStrings["constr"].ConnectionString;
        using (SqlConnection con = new SqlConnection(constr))
        {
            using (SqlCommand cmd = new SqlCommand("SELECT * FROM Customers", con))
            {
                using (SqlDataAdapter da = new SqlDataAdapter(cmd))
                {
                    DataTable dt = new DataTable();
                    da.Fill(dt);
                    gvCustomers.DataSource = dt;
                    gvCustomers.DataBind();
                }
            }
        }
    }
}
protected void gvCustomers_onpageindexchanging(object sender, GridViewPageEventArgs e)
{
    gvCustomers.PageIndex = e.NewPageIndex;
    BindGrid();
}
protected void Change(object sender, EventArgs e)
{
    BindGrid();
}
public void BindGrid()
{
    string country = DropDownList1.SelectedItem.Text;
    string constr = ConfigurationManager.ConnectionStrings["constr"].ConnectionString;
    using (SqlConnection con = new SqlConnection(constr))
    {
        using (SqlCommand cmd = new SqlCommand("SELECT * FROM Customers WHERE Country = @country", con))
        {
            cmd.Parameters.AddWithValue("@country", country);
            using (SqlDataAdapter da = new SqlDataAdapter(cmd))
            {
                DataTable dt = new DataTable();
                da.Fill(dt);
                gvCustomers.DataSource = dt;
                gvCustomers.DataBind();
            }
        }
    }
}
VB.Net
Protected Sub Page_Load(ByVal sender As Object, ByVal e As EventArgs) Handles Me.Load
    If Not Me.IsPostBack Then
        Dim constr As String = ConfigurationManager.ConnectionStrings("constr").ConnectionString
        Using con As SqlConnection = New SqlConnection(constr)
            Using cmd As SqlCommand = New SqlCommand("SELECT * FROM Customers", con)
                Using da As SqlDataAdapter = New SqlDataAdapter(cmd)
                    Dim dt As DataTable = New DataTable()
                    da.Fill(dt)
                    gvCustomers.DataSource = dt
                    gvCustomers.DataBind()
                End Using
            End Using
        End Using
    End If
End Sub
Protected Sub gvCustomers_onpageindexchanging(ByVal sender As Object, ByVal e As GridViewPageEventArgs)
    gvCustomers.PageIndex = e.NewPageIndex
    BindGrid()
End Sub
Protected Sub Change(ByVal sender As Object, ByVal e As EventArgs)
    BindGrid()
End Sub
Public Sub BindGrid()
    Dim country As String = DropDownList1.SelectedItem.Text
    Dim constr As String = ConfigurationManager.ConnectionStrings("constr").ConnectionString
    Using con As SqlConnection = New SqlConnection(constr)
        Using cmd As SqlCommand = New SqlCommand("SELECT * FROM Customers WHERE Country = @country", con)
            cmd.Parameters.AddWithValue("@country", country)
            Using da As SqlDataAdapter = New SqlDataAdapter(cmd)
                Dim dt As DataTable = New DataTable()
                da.Fill(dt)
                gvCustomers.DataSource = dt
                gvCustomers.DataBind()
            End Using
        End Using
    End Using
End Sub
Screenshot
