Update iMac

Yes a Mid 2011 iMac can be updated from Mountain Lion 10.9 to macOS High Sierra 10.13.

see > How to get old versions of macOS - Apple Support

+1, without know the exact model year (look in About this Mac when clicking the symbol in the upper left of the display) we cannot answer your question. Once you know the model year, you can also Google "System requirements for Mountain Lion" and find the same information we will find.

Also, please remember Mountain Lion is a very old version of Mac OS, if you need to run more contemporary versions of Mac OS please consider replacing the computer with a more up-to-date Mac.

Yes, you can "upgrade" it to ML however somehow you will need to find the original upgrade DVD. Apple discontinued it years ago so your best bet is to find one used, however if you look on E-Bay you may find it. DO NOT buy a counterfeit copy, when you find the original DVD contact the seller and get them to guarantee their DVD is a genuine Apple DVD. If they say it is not, but it will work then start looking again.

As stated you have a VERY VERY old computer, please consider replacing the machine if you have a need to run more current versions of Mac OS. We run into similar users as yourself almost daily, the longer you hold onto old technology the more difficult it is to maintain them. Essentially you are painting yourself into a corner by continuing on with your 2011.

No prob. plus you make a good point because High Sierra and Safari 13 are no longer supported.

For example: I have a couple of old Mac Mini's (2010 & 2011) setup as media servers and have to use Mozilla FireFox to access Xfinity Live TV because High Sierra 10.13 and Safari 13 are too old.

Depends on exactly what model Mac it is. What Model and year does it say when you click on the  menu and “About this Mac”?

I stand corrected, I was thinking of Snow Leopard rather than Mountain lion. I'm cat confused!