I'm looking at Azure Dev Ops webhooks triggers and looks like there's one only for when a build was completed. Is there any way to be notified when a build was started? Since a job can stay in queue for a while. So I'd need to know when it's in queue, when it's started and completed.
No. There is no Webhook trigger for Build Start type.
Webhook is mainly to trigger another job once TFS registers an event. If you want to execute something on build start you could just have that task in the build itself. If you want to know the build start time and end time, you could get this info from the build details or through API.
If you are looking to notify yourself when build starts, you could have a PowerShell task inside the build to use the API and send the details.