The reason all data is being exported is
foreach (GridViewRow row in GridView1.Rows)
Also, you need to change the line
btn_exportToexcel.Click += new System.EventHandler(btn_exportToexcel_Clicked);
btn_exportToexcel.Command += new System.CommandEventHandler(btn_exportToexcel_Clicked)
Then assign a commandargument value to btn_exportToExcel. Set the command argument for the export button to the unique key field value of the child gridview row. You might need to do this inside protected void Child_gv_RowDataBound.
Then change the private void btn_exportToexcel_Clicked to private void btn_exportToexcel_Command(object sender, CommandEventArgs e) and you have the child row id as the 'e' value.
Inside this method, get rid of the for each row code and just export the child row data found using the 'e' value.