using System.Windows.Forms

public Form1()
{
              //здесь обычная связка dataGridView1 bindingSource1
            dataGridView1.DataSource = bindingSource1;
 
            var query = from c in db.Dogovors
                                         join mo in db.Orgs on c.dg_mo_id equals mo.mo_id
                                         select new
                                       {
                                           c.id,
                                           c.num,
                                           c.descr,
                                           mo.аullname,
                                        };
          bindingSource1.DataSource = query;

}
 

private void toolStripButton1_Click(object sender, EventArgs e)
        {
            var cur = bindingSource1.Current;
// ЧТО ЖЕ здесь надо написать что бы узнать чему павно поле id
.......
}

public class Dogovor
    {
        public int dg_id;
        public string num;
        public string descr;
        public string fаullname;
}
 var query = from c in db.Dogovors
                                         join mo in db.Orgs on c.dg_mo_id equals mo.mo_id
                                         select new Dogovor
                                       {
                                           id = c.id,
                                           num = c.num,
                                          descr= c.descr,
                                          fаullname= mo.fаullname
                                        };
          bindingSource1.DataSource = query;

public class Dogovor
    {
        public int dg_id { get; set; }
        public string num { get; set; }
        public string descr { get; set; }
        public string fаullname { get; set; }
}