Please refer this code
Gridview inside Gridview in ASP.Net using C# and VB.Net
And refer this code for showing the Modal Popup in child GridView.
In the Above link ButtonField is use to show the modal popup.
<asp:ButtonField Text="Select" CommandName="Select" />
If you want to Use LinkButton with OnClick event then please add this Html code in Child GridView Columns
<asp:TemplateField>
<ItemTemplate>
<asp:LinkButton Text="Select" runat="server" OnClick="SelectRow" />
</ItemTemplate>
</asp:TemplateField>
Oclick event of LinkButton to show the Modal popup.
protected void ExportExcel(object sender, EventArgs e)
{
GridViewRow row = (sender as LinkButton).NamingContainer as GridViewRow;
lblId.Text = row.Cells[0].Text;
lblName.Text = row.Cells[1].Text;
lblCountry.Text = (row.FindControl("lblCountry") as Label).Text;
mpe.Show();
}