You need to set the property Iconlike this example (taken from the Mahapps demo application):
<Controls:MetroWindow x:Class="MetroDemo.MainWindow"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
xmlns:Controls="clr-namespace:MahApps.Metro.Controls;assembly=MahApps.Metro"
Title="MahApps.Metro - Demo Application"
Width="960" Height="600"
Icon="mahapps.metro.logo2.ico"
ShowIconOnTitleBar="True"
ShowTitleBar="True">
<Grid />
</Controls:MetroWindow>
Mahapps, ( , -), . GitHub