Hi mvadukul,
I have created sample code which fulfill your requirement.
HTML
<form id="form1" runat="server">
<div>
    <br />
    <br />
    <center>
        <asp:GridView ID="gvEmployees" AutoGenerateColumns="false" runat="server" OnRowDataBound="gvEmployees_RowDataBound">
            <Columns>
                <asp:TemplateField ItemStyle-HorizontalAlign="Center" HeaderText="Flag">
                    <ItemTemplate>
                        <div>
                            <asp:Label ID="lblFlagName" Text='<%#Eval("FlagColor") %>' Visible="false" runat="server" />
                            <ul class="dropdown">
                                <a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-haspopup="true"
                                    aria-expanded="false">
                                    <asp:Label ID="lblflag" class="fa fa-flag fa-1x" runat="server" />
                                    <span class="caret"></span></a>
                                <ul class="dropdown-menu" style="min-width: 50px; left: -1; min-height: 120px; padding: 20px 0;
                                    margin: 4px 0 0; border: 3px solid rgba(6, 172, 179, 0.15);">
                                    <center>
                                        <li>
                                            <asp:LinkButton ID="btnBlack" OnClick="OnClick" CommandName="Black" runat="server"><i style="color: black" class="fa fa-flag fa-1x"></i>
                                            </asp:LinkButton>
                                        </li>
                                        <li>
                                            <asp:LinkButton ID="btnRed" OnClick="OnClick" CommandName="Red" runat="server"> <i style="color: Red" class="fa fa-flag fa-1x"></i>
                                            </asp:LinkButton>
                                        </li>
                                        <li>
                                            <asp:LinkButton ID="btnGreen" OnClick="OnClick" CommandName="Green" runat="server">
                                        <i style="color: Green" class="fa fa-flag fa-1x"></i>
                                            </asp:LinkButton>
                                        </li>
                                        <li>
                                            <asp:LinkButton ID="btnYellow" OnClick="OnClick" CommandName="Yellow" runat="server">
                                        <i style="color: Yellow" class="fa fa-flag fa-1x"></i>
                                            </asp:LinkButton>
                                        </li>
                                    </center>
                                </ul>
                            </ul>
                        </div>
                    </ItemTemplate>
                </asp:TemplateField>
                <asp:TemplateField HeaderText="Customer Details">
                    <ItemTemplate>
                        <asp:Label ID="lblId" Text='<% #Eval("Id")%>' runat="server" />
                        <asp:Label ID="lblEmployeeName" Text='<% #Eval("Name")%>' runat="server" />
                        <asp:Label ID="Label1" Text='<%#Eval("Country")%>' runat="server" />
                    </ItemTemplate>
                </asp:TemplateField>
            </Columns>
        </asp:GridView>
    </center>
    <div>
        <script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.9.1/jquery.min.js"></script>
        <script type="text/javascript" src="http://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js"></script>
        <link rel="stylesheet" href="http://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css" />
        <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css" />
        <style type="text/css">
            body
            {
                font-family: Arial;
                font-size: 10pt;
            }
            table
            {
                border: 1px solid #ccc;
            }
            table th
            {
                background-color: #F7F7F7;
                color: #333;
                font-weight: bold;
            }
            table th, table td
            {
                padding: 5px;
                border-color: #ccc;
            }
        </style>
    </div>
</form>
C#
private string config = ConfigurationManager.ConnectionStrings["Sqlconn"].ToString();
protected void Page_Load(object sender, EventArgs e)
{
    if (!IsPostBack)
    {
        this.GetData();
    }
}
private void GetData()
{
    SqlConnection con = new SqlConnection(config);
    SqlCommand cmd = new SqlCommand("Select * From CustomersTest", con);
    SqlDataAdapter sda = new SqlDataAdapter(cmd);
    DataTable dt = new DataTable();
    sda.Fill(dt);
    gvEmployees.DataSource = dt;
    gvEmployees.DataBind();
}
protected void OnClick(object sender, EventArgs e)
{
    string colorName = (sender as LinkButton).CommandName;
    GridViewRow row = (sender as LinkButton).NamingContainer as GridViewRow;
    int Id = Convert.ToInt32((row.FindControl("lblId") as Label).Text);
    SqlConnection con = new SqlConnection(config);
    SqlCommand cmd = new SqlCommand("Update CustomersTest Set FlagColor=@FlagColor WHERE Id=@Id", con);
    cmd.Parameters.AddWithValue("@FlagColor", colorName);
    cmd.Parameters.AddWithValue("@Id", Id);
    con.Open();
    cmd.ExecuteNonQuery();
    con.Close();
    this.GetData();
}
protected void gvEmployees_RowDataBound(object sender, GridViewRowEventArgs e)
{
    if (e.Row.RowType == DataControlRowType.DataRow)
    {
        string colorName = (e.Row.FindControl("lblFlagName") as Label).Text;
        (e.Row.FindControl("lblflag") as Label).Style.Add("color", colorName);
    }
}
Vb.net
Private config As String = ConfigurationManager.ConnectionStrings("Sqlconn").ToString()
Protected Sub Page_Load(sender As Object, e As EventArgs)Handles Me.Load
	If Not IsPostBack Then
		Me.GetData()
	End If
End Sub
Private Sub GetData()
	Dim con As New SqlConnection(config)
	Dim cmd As New SqlCommand("Select * From CustomersTest", con)
	Dim sda As New SqlDataAdapter(cmd)
	Dim dt As New DataTable()
	sda.Fill(dt)
	gvEmployees.DataSource = dt
	gvEmployees.DataBind()
End Sub
Protected Sub OnClick(sender As Object, e As EventArgs)
	Dim colorName As String = TryCast(sender, LinkButton).CommandName
	Dim row As GridViewRow = TryCast(TryCast(sender, LinkButton).NamingContainer, GridViewRow)
	Dim Id As Integer = Convert.ToInt32(TryCast(row.FindControl("lblId"), Label).Text)
	Dim con As New SqlConnection(config)
	Dim cmd As New SqlCommand("Update CustomersTest Set FlagColor=@FlagColor WHERE Id=@Id", con)
	cmd.Parameters.AddWithValue("@FlagColor", colorName)
	cmd.Parameters.AddWithValue("@Id", Id)
	con.Open()
	cmd.ExecuteNonQuery()
	con.Close()
	Me.GetData()
End Sub
Protected Sub gvEmployees_RowDataBound(sender As Object, e As GridViewRowEventArgs)
	If e.Row.RowType = DataControlRowType.DataRow Then
		Dim colorName As String = TryCast(e.Row.FindControl("lblFlagName"), Label).Text
		TryCast(e.Row.FindControl("lblflag"), Label).Style.Add("color", colorName)
	End If
End Sub
Sql
CREATE TABLE CustomersTest
( Id INT
 ,Name VARCHAR(50)
 ,Country VARCHAR(50)
 ,FlagColor VARCHAR(20)	
);
Insert into CustomersTest Values(1,'Mudassar Khan','India','Black');
Insert into CustomersTest Values(2,'Maria','Austria','Black');
Insert into CustomersTest Values(3,'Christina Berglund','Austria','Black');
Insert into CustomersTest Values(4,'Thomas Hardy','Brazil','Black');
Insert into CustomersTest Values(5,'Ana Trujillo','UK','Black');
Screenshot