Hi Worthington, Firefox normally updates itself between Firefox sessions without asking you to download anything. You are correct to be suspicious of pages or popups promoting updates.

You can set Firefox to inform you of updates before downloading them, and those notifications will appear as a "door hanger" from the menu button. This picture is from Firefox 56, but it's largely the same now:

To manually trigger an update, use the method in this article: Update Firefox to the latest release. Your Firefox identified itself as version 58, and the current version is 60.

If you get a pop-up message asking to update Firefox or plugins or scanning for malware then such a message is likely a scam and you should NEVER respond to such an alert to avoid getting infected with malware.

• only update Firefox via "Help -> About" or by downloading and installing Firefox from the Mozilla server and never via a pop-up or link on a web page.
• plugins should only be updated via the plugin itself or by visiting the home page of the plugin.

See also:

• https://support.mozilla.org/en-US/kb/i-found-fake-firefox-update