A Hyperlink is an inline, not a control, so it lives inside a TextBlock and is styled rather than templated. MahApps gives it the Metro treatment: underlined, in the ordinary text colour, and only coloured once the pointer is over it.

Note the first panel. A MahApps hyperlink is not blue at rest — it takes MahApps.Brushes.Text, the same colour as the words around it, and relies on the underline to mark it as a link. The accent-ish colour is the hover state. That is a deliberate part of the look, but it surprises people expecting the browser convention.
The implicit style
Styles/Controls.xaml applies MahApps.Styles.Hyperlink to every Hyperlink. Merging that dictionary — which the quick start does — is all it takes:
<ResourceDictionary Source="pack://application:,,,/MahApps.Metro;component/Styles/Controls.xaml" />
<TextBlock>
The source code is <Hyperlink NavigateUri="https://github.com/MahApps/MahApps.Metro">hosted on GitHub</Hyperlink>
and includes everything needed to build it yourself.
</TextBlock>

What the style does
MahApps.Styles.Hyperlink is based on WPF's own default hyperlink style rather than replacing it, and adds two setters and three triggers:
Foreground |
MahApps.Brushes.Text |
TextDecorations |
Underline |
on IsMouseOver |
Foreground becomes MahApps.Brushes.Highlight |
on IsEnabled="False" |
Foreground becomes the system grey-text colour |
on IsEnabled="True" |
Cursor becomes Hand, with ForceCursor="True" |
ForceCursor is the one worth knowing about: it makes the hand cursor win over whatever the surrounding control would otherwise show, so a link inside a control with its own cursor still looks clickable.
To make links read as links in the browser sense, override the foreground — either on the link or through a style of your own:
<Style BasedOn="{StaticResource MahApps.Styles.Hyperlink}" TargetType="{x:Type Hyperlink}">
<Setter Property="Foreground" Value="{DynamicResource MahApps.Brushes.Accent}" />
</Style>
Keep the BasedOn, or the underline, the hover colour and the hand cursor go with it.
Navigating
The style does nothing about navigation — that is WPF's job, and WPF does not open a browser for you. Handle RequestNavigate:
private void OnRequestNavigate(object sender, RequestNavigateEventArgs e)
{
Process.Start(new ProcessStartInfo(e.Uri.AbsoluteUri) { UseShellExecute = true });
e.Handled = true;
}
<Hyperlink NavigateUri="https://mahapps.com" RequestNavigate="OnRequestNavigate">mahapps.com</Hyperlink>
In a DataGrid
MahApps.Styles.Hyperlink.DataGrid is what a DataGridHyperlinkColumn uses for its cells. Despite the name it targets TextBlock, not Hyperlink: a hyperlink column puts a TextBlock in the cell and the link inside it, so the style dresses the block and carries a nested Hyperlink style in its Style.Resources for the link itself.
You do not need to set it — the DataGrid style wires it up through DataGridHelper, as it does for every other column type. See DataGrid Columns.