Only when the team has a spend limit. In Team Settings, choose the Budget Alert Threshold: 50%, 80% (the default) or 90% of the team spend limit. When the team's spending reaches it, every team admin gets one email, "Spend limit alert" with the team's name, showing the spending so far and the limit. The organization owner is admin of each team they create.
Sent once: it isn't sent again until someone clicks Save Changes in Team Settings, even with the limit unchanged. That re-arms it, so if spending is still past the threshold, it's sent again.
Timing: the check runs in the background, so the email can arrive a few minutes after the threshold is crossed.
Member limits set on the Members page don't send alerts.
Spend limits count all-time spending, not monthly. See How do organizations and teams work? and Spend Limits.