Can I update OS X Mountain Lion

Unless you haven't updated your computer in many years there have been no recent updates for Mountain Lion.

If you mean upgrade, it is hard to give advice when we know absolutely nothing about your computer model or year. The various ways of checking for software may only show upDATES to the current main system version you are running (e.g., 10.4.8 --> 10.4.11), not upGRADES to a higher system version such as a jump from El Capitan (10.11) to Catalina (10.15). Note the difference between an update and an upgrade.

Upgrading a macOS depends upon the model Mac, specifications, and year. To get more information about your computer, choose:  (Apple menu) > 'About This Mac' in the upper left corner of any window, then "More Info..." or "System Report". There's more about this in "About System Information [Profiler] on your Mac" - System Information User Guide for Mac - Apple Support

Use the computer information, including the model year, and the links below to determine what is possible with your computer.

Click on this link to read how to get different versions of macOS --> How to download macOS - Apple Support

- The article also has links to system requirements for each OS version which you should check against your exact computer model. Note some of the model identifier articles are outdated. For any model produced in the last 5 years or so you should use the system requirements links for each OS version instead. This web site also has a reliable presentation of model and OS version compatibility: https://eshop.macsales.com/guides/Mac_OS_X_Compatibility

- It helps to have more RAM (memory) than what is noted as "minimum" for any particular system or your computer may run slowly.

- You may find you must use Safari to get links for High Sierra and later using the Mac App Store application to open to a functioning page.

- Sierra, El Capitan, and Yosemite download as a .pkg. Click on this to have a converted version placed in your Applications folder as an "Install OS..." application. Run the installer and follow the directions.

- Upgrades may take some time (up to an hour or more) to install, with restarts and blank screens.

Before upgrading, back up at least your data in case something goes wrong. See this support document: How to back up your Mac - How to back up your Mac - Official Apple Support

Check if your old apps will still work with with the OS you want to install - https://roaringapps.com/apps This is particularly critical if you are upgrading to Catalina or newer, as older 32 bit apps will no longer work. To check for those, try this 32 bit application scanner application --> https://www.stclairsoft.com/Go64/

If an error occurred while updating or installing macOS --> If an error occurred while updating or installing macOS - Apple Support