Updates for Adobe Flash come thru the Flash updater, not via Add-on updates.

Go the Windows Control Panel > Flash Player -> Flash Player Settings Manager and see the Updates tab and use the Check Now button.

BTW, flash is a plugin. Not an add-on. That is why it did not update.

FredMcD said

BTW, flash is a plugin. Not an add-on. That is why it did not update.

A Plugin is a Addon as otherwise it would not be listed in the Addons Manager. The word Addons is a general term that started back during Firefox 2.0/3.0 days. https://addons.mozilla.org/faq

scheung said

When I click on the Update add-ons button, my Flash plug-in (ver 29.0.0.113) does not show that there is a new version even though I have checked on Adobe's website that there is a newer version available (29.0.0.140). Firefox version: 59.0.2 (64-bit)

NPAPI Plugins that Firefox makes use of (only Flash Player from Adobe now days) is installed on system and not in Firefox like say Extensions are.

Type about:plugins with a : in the Location (address) to see where the Flash Player Plugin is located on your system.

To keep Flash updated on Windows you either use the Flash updater or by download from Adobe and install it after uninstalling old version.