You shoud be using Single Quotes, incase if you are doing this for TSQL.
If you want to make your Double Quotes logic working then, do as below.
1) Use @ before your string which has Double Quote in it.
2) Use StringBuilder which will give you better performance as compare to regular string operations.
3) Make sure, you want to show messageBox inside "foreach" loop or outside.
System.Text.StringBuilder addquery = new System.Text.StringBuilder();
string fieldname = string.Empty;
foreach (DataRow dr in dt.Rows)
{
fieldname = dr["FIELDNAME"].ToString().Trim();
addquery.Append(@"""");
addquery.Append(fieldname);
addquery.Append(@""",");
messageBox.Show(addquery);
}