Add Postback trigger for Button1 as mentioned below :
<asp:UpdatePanel ID="C_Compte" runat="server" UpdateMode="Conditional">
<ContentTemplate>
Métier:
<asp:TextBox ID="tbAddMetier" CssClass="Textboxes" Width="300px" Height="25px" runat="server"></asp:TextBox>
<asp:Button ID="btnAddMetier" runat="server" Text="Ajouter" CssClass="OffreEmploiSearch" OnClick="Button1_Click" />
<h4 style="font-weight: bold">Mes métiers</h4>
<asp:Repeater ID="rptrMetier" runat="server">
<ItemTemplate>
<div style="font: normal normal normal 14px/1.4em play, sans-serif; color: rgb(95, 94, 93); padding-left: 10px; line-height: 22px">
<strong><%# Eval("Lebelle") %></strong></a>
<asp:ImageButton ID="ImageButton1" runat="server" ImageUrl="~/Design_Ressources/img/attachment.png" OnCommand="ImageButton1_Command" CommandArgument='<%# Eval("Lebelle") %>' />
</div>
</ItemTemplate>
</asp:Repeater>
</ContentTemplate>
<Triggers>
<asp:PostBackTrigger ControlID="btnAddMetier"/>
</Triggers>
</asp:UpdatePanel>