Hi Sowmya90,

Check with the below code. Paging is working.

HTML

<script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
<script type="text/javascript">
    $(function () {
        $('[id*=Grd_Dtls] tr td').on('click', function () {
            if ($(this).index() > 1) {
                $('[id*=Grd_Dtls] tr td').each(function () {
                    if ($(this).index() > 1) {
                        $(this).find('span').show();
                        $(this).find('input[type=text]').hide();
                        $(this).find('[id*=ddl_List2]').hide();
                        $(this).find('span').html($(this).find('input[type=text]').val());
                        $(this).find('span').html($(this).find('[id*=ddl_List2]').val());
                    }
                });
                var val = $(this).find('span').html();
                $(this).find('span').hide();
                $(this).find('[id*=ddl_List2]').val(val);
                $(this).find('[id*=ddl_List2]').show();
                $(this).find('[id*=ddl_List2]').focus();
                $(this).find('input[type=text]').val(val);
                $(this).find('input[type=text]').show();
                $(this).find('input[type=text]').focus();
                $('[id*=ddl_List2]').on('click', function () {
                    return false;
                });
            }
        });
    });
</script>
<asp:GridView ID="Grd_Dtls" runat="server" AutoGenerateColumns="False" PageSize="2"
    AllowPaging="true" OnPageIndexChanging="Grd_Dtls_PageIndexChanging">
    <Columns>
        <asp:BoundField DataField="ID" HeaderText="ID" />
        <asp:BoundField DataField="name" HeaderText="Name" />
        <asp:TemplateField HeaderText="Text1">
            <ItemTemplate>
                <asp:Label ID="lbl_Text1" runat="server" Text='<%# Bind("Text1") %>'></asp:Label>
                <asp:TextBox ID="txt_Text1" runat="server" Text='<%# Bind("Text1") %>' CssClass="hideControl"></asp:TextBox>
            </ItemTemplate>
        </asp:TemplateField>
        <asp:TemplateField HeaderText="List2">
            <ItemTemplate>
                <asp:Label ID="lbl_List2" runat="server" Text='<%# Bind("List2") %>'></asp:Label>
                <asp:DropDownList runat="server" ID="ddl_List2" CssClass="hideControl" SelectedValue='<%# Bind("List2") %>'>
                    <asp:ListItem Value="0">0</asp:ListItem>
                    <asp:ListItem Value="1">1</asp:ListItem>
                </asp:DropDownList>
            </ItemTemplate>
        </asp:TemplateField>
    </Columns>
</asp:GridView>

CSS

<style type="text/css">
    .hideControl
    {
        display: none;
    }
</style>

Code

protected void Page_Load(object sender, EventArgs e)
{
    if (!this.IsPostBack)
    {
        BindGrid();
    }
}

private void BindGrid()
{
    System.Data.DataTable dt = new System.Data.DataTable();
    dt.Columns.AddRange(new System.Data.DataColumn[] { 
        new System.Data.DataColumn("Id"), new System.Data.DataColumn("Name"), 
        new System.Data.DataColumn("Text1"), new System.Data.DataColumn("List2") });
    dt.Rows.Add(1, "John Hammond", "United States", 0);
    dt.Rows.Add(2, "Mudassar Khan", "India", 1);
    dt.Rows.Add(3, "Suzanne Mathews", "France", 1);
    dt.Rows.Add(4, "Robert Schidner", "Russia", 0);
    this.Grd_Dtls.DataSource = dt;
    this.Grd_Dtls.DataBind();
}

Screenshot