If you don't want the page to post back, you'll need to use Javascript to change the visibility of the buttons.
Something like this should work:
Code behind:
protected void Page_Load(object sender, EventArgs e)
{
}
Markup:
<asp:Button id="Button1" runat="server" class="btn btn-primary btn-lg" OnClientClick="button1Click(); return false;" Text="11:00AM" style= "font-family: Arial, Helvetica, sans-serif;" />
<asp:button id="Button2" runat="server" class="btn btn-outline-info d-none" OnClick="Button2_Click" Text="Confirm"/>
<asp:button id="Button3" runat="server" class="btn btn-outline-info d-none" OnClick="Button3_Click" Text="11:00AM"/>
Javascript:
function button1Click(){
$(this).addClass("d-none");
$("button[id$='Button2'], button[id$='Button3']").removeClass("d-none");
}