Hi Vanessa,

Refer below sample code.

HTML

Home

<asp:Button ID="btnShowPopup" runat="server" Text="Show Popup" />
<div id="dialog" style="display: none">
    <asp:GridView ID="GridView1" runat="server" AutoGenerateColumns="False" ShowHeaderWhenEmpty="True"
        OnSelectedIndexChanged="OnSelectedIndexChanged" OnPageIndexChanging="OnPageIndexChanging"
        Width="1500px" Height="188px" PageSize="3" AllowPaging="true" DataKeyNames="PENTransCode">
        <AlternatingRowStyle Font-Size="Small" BackColor="#CCCCCC" />
        <Columns>
            <asp:TemplateField>
                <ItemTemplate>
                    <asp:LinkButton ID="lnkDetails" runat="server" Text="Edit"
                        PostBackUrl='<%# "~/Default.aspx?RowIndex=" + Container.DataItemIndex %>'></asp:LinkButton>
                </ItemTemplate>
            </asp:TemplateField>
            <asp:BoundField ItemStyle-Width="500px" DataField="PENTransCode" HeaderText="PENTransCode">
                <HeaderStyle ForeColor="#3366CC" />
                <ItemStyle Width="500px" />
            </asp:BoundField>
            <asp:TemplateField HeaderText="NameofService">
                <ItemTemplate>
                    <asp:Label ID="lbNameofService" runat="server" Text='<%# Eval("NameofService") %>' />
                </ItemTemplate>
            </asp:TemplateField>
            <asp:TemplateField>
                <ItemTemplate>
                    <asp:Button ID="lnkDownload" runat="server" Text="Download" OnClick="DownloadFile"></asp:Button>
                </ItemTemplate>
            </asp:TemplateField>
        </Columns>
        <EditRowStyle Font-Size="Small" />
        <HeaderStyle ForeColor="#996633" Font-Size="Small" BackColor="#FFCC66"
            BorderColor="#666666" BorderWidth="1px" Height="30px"
            HorizontalAlign="Center" VerticalAlign="Middle" />
        <RowStyle Font-Size="Small" BackColor="White" />
    </asp:GridView>
</div>

<script type="text/javascript" src="https://code.jquery.com/jquery-1.12.4.js"></script>
<link rel="stylesheet" href="https://code.jquery.com/ui/1.12.1/themes/base/jquery-ui.css" />
<script type="text/javascript" src="https://code.jquery.com/ui/1.12.1/jquery-ui.js"></script>
<script type="text/javascript">
    $(function () {
        $("[id*=btnShowPopup]").click(function () {
            ShowPopup();
            return false;
        });
    });

    function ShowPopup() {
        $("#dialog").dialog({
            title: "Returned Accomplishment Reports",
            width: 250,
            buttons: {
                Ok: function () {
                    $(this).dialog('close');
                }
            },
            modal: true
        });
    }
</script>

Default

<asp:TextBox ID="txtPENTransCode" runat="server" />
<asp:Label ID="lbNameofService" runat="server" />

Code

Home

C#

protected void Page_Load(object sender, EventArgs e)
{
    if (!this.IsPostBack)
    {
        BindGrid();
    }
}

private void BindGrid()
{
    DataTable dt = new DataTable();
    dt.Columns.AddRange(new DataColumn[]
    {
        new DataColumn("PENTransCode"),
        new DataColumn("NameofService")
    });
    dt.Rows.Add(1, "Service 1");
    dt.Rows.Add(2, "Service 2");
    dt.Rows.Add(3, "Service 3");
    dt.Rows.Add(4, "Service 4");
    dt.Rows.Add(5, "Service 5");
    GridView1.DataSource = dt;
    GridView1.DataBind();
}

protected void OnSelectedIndexChanged(object sender, EventArgs e)
{

}

protected void DownloadFile(object sender, EventArgs e)
{
    Button lnkbtn = sender as Button;
    GridViewRow gvrow = lnkbtn.NamingContainer as GridViewRow;
    string filePath = GridView1.DataKeys[gvrow.RowIndex].Value.ToString();
    Response.ContentType = ContentType;
    Response.AddHeader("Content-Disposition", "attachment;filename=\"" + filePath + "\"");
    Response.TransmitFile(Server.MapPath(filePath));
    Response.End();
}

protected void OnPageIndexChanging(object sender, GridViewPageEventArgs e)
{
    GridView1.PageIndex = e.NewPageIndex;
    BindGrid();
    ClientScript.RegisterStartupScript(this.GetType(), "Popup", "ShowPopup();", true);
}

VB.Net

Protected Sub Page_Load(ByVal sender As Object, ByVal e As EventArgs) Handles Me.Load
    If Not Me.IsPostBack Then
        BindGrid()
    End If
End Sub

Private Sub BindGrid()
    Dim dt As DataTable = New DataTable()
    dt.Columns.AddRange(New DataColumn() {
                        New DataColumn("PENTransCode"),
                        New DataColumn("NameofService")})
    dt.Rows.Add(1, "Service 1")
    dt.Rows.Add(2, "Service 2")
    dt.Rows.Add(3, "Service 3")
    dt.Rows.Add(4, "Service 4")
    dt.Rows.Add(5, "Service 5")
    GridView1.DataSource = dt
    GridView1.DataBind()
End Sub

Protected Sub OnSelectedIndexChanged(ByVal sender As Object, ByVal e As EventArgs)
End Sub

Protected Sub DownloadFile(ByVal sender As Object, ByVal e As EventArgs)
    Dim lnkbtn As Button = TryCast(sender, Button)
    Dim gvrow As GridViewRow = TryCast(lnkbtn.NamingContainer, GridViewRow)
    Dim filePath As String = GridView1.DataKeys(gvrow.RowIndex).Value.ToString
    Response.ContentType = ContentType
    Response.AddHeader("Content-Disposition", "attachment;filename=" & filePath)
    Response.TransmitFile(Server.MapPath(filePath))
    Response.End()
End Sub

Protected Sub OnPageIndexChanging(ByVal sender As Object, ByVal e As GridViewPageEventArgs)
    GridView1.PageIndex = e.NewPageIndex
    BindGrid()
End Sub

Default

C#

protected void Page_Load(object sender, EventArgs e)
{
    if (this.Page.PreviousPage != null)
    {
        int rowIndex = int.Parse(Request.QueryString["RowIndex"]);
        GridView GridView1 = (GridView)this.Page.PreviousPage.FindControl("GridView1");
        GridViewRow row = GridView1.Rows[rowIndex % GridView1.PageSize];
        txtPENTransCode.Text = row.Cells[1].Text;
        lbNameofService.Text = ((Label)row.FindControl("lbNameofService")).Text;
    }
}

VB.Net

Protected Sub Page_Load(ByVal sender As Object, ByVal e As EventArgs) Handles Me.Load
    If Me.Page.PreviousPage IsNot Nothing Then
        Dim rowIndex As Integer = Integer.Parse(Request.QueryString("RowIndex"))
        Dim GridView1 As GridView = DirectCast(Me.Page.PreviousPage.FindControl("GridView1"), GridView)
        Dim row As GridViewRow = GridView1.Rows(rowIndex Mod GridView1.PageSize)
        txtPENTransCode.Text = row.Cells(1).Text
        lbNameofService.Text = CType(row.FindControl("lbNameofService"), Label).Text
    End If
End Sub

Screenshot