A message is in Gmail, but no desktop alert appeared. First establish whether the message arrived; delivery and notification are different checks. For desktop alerts, Google documents signing in and keeping Gmail open in a supported browser such as Chrome, Firefox or Safari.
Check the Gmail choice
On a computer, open Settings → See all settings and find Desktop notifications. Choose new-mail notifications, important-mail notifications or off, then save changes. “Important mail” is a narrower choice, not a promise to alert for every arrival.
There is another scope limit: with inbox categories enabled, new-mail notifications apply to messages in Primary. For a harmless test, inspect whether the expected message arrived in Primary or another category. An alert missing for a Promotions message does not by itself show failed mail delivery.
Check the browser before changing mail rules
Gmail cannot show the intended alerts if browser notifications are blocked. Check the browser’s notification permission and the computer’s relevant notification settings. Do not change spam filters, forwarding or account access merely to diagnose an absent alert.
If you want a sound, check the separate mail notification sound choice; None deliberately disables that sound. Keep the visual alert and sound tests separate. Use the intended account and browser profile, then observe the next permitted test message with Gmail open. No message was sent or browser/account setting changed for this article; phone notifications follow separate device-specific instructions.