In Prometheus land everything hangs on labels: You use labels in a query, you use labels for alert routing & you use labels for silences.
So in your query you would include a label selector that excludes the server(s) you want. This could be on an instance label (which might have the IP address or DNS name), but could equally be a “type” label or something else.
Overall it really depends on what you are wanting to do. Changing the alert expression would make an alert never fire at all, while adjusting the alert routing or creating a silence will still cause the alert to fire, but it just wouldn’t have any effect. However that does mean that going to the Alertmanager or Prometheus UI would list that alert as firing (because it is).
As an example of a query which excludes something you could do up{instance!="ignore.example.com"} == 0 which would fire for all servers where up = 0, except for the one with the instance label of ignore.example.com