Add trigger in second update panel for control from first update panel for which you want to update second update panel.
Try this solution with removing trigger & you will get idea.
<asp:UpdatePanel ID="UpdatePanel1" runat="server" UpdateMode="Conditional">
<ContentTemplate>
<asp:Label ID="lbl1" runat="server" Text="lbl1"></asp:Label>
<asp:Button ID="btn" runat="server" onclick="btn_Click" Text="Click" />
</ContentTemplate>
</asp:UpdatePanel>
<asp:UpdatePanel ID="UpdatePanel2" runat="server" UpdateMode="Conditional">
<Triggers>
<asp:PostBackTrigger ControlID="btn" />
</Triggers>
<ContentTemplate>
<asp:Label ID="lbl2" runat="server" Text="lbl2"></asp:Label>
</ContentTemplate>
</asp:UpdatePanel>
protected void btn_Click(object sender, EventArgs e)
{
lbl1.Text = "change1";
lbl2.Text = "change2";
}