Is it possible to see the content of game updates on Steam?

Sometimes. It depends on if the game has released information about what has been updated. Some games are really good about it, others are not.

To view where it would be, go to your downloads page (Library > Downloads) and where the download is, you should see a view news link on the right hand side of the page. This takes you to the Steam News page for that game, and is where games would be able to share what has changed with recent updates. If there is a lot of news and you just want the changes, you can filter on the Product Updates category to see only items that notes changes made.

Note some games may use an external site or system for reporting this information, but this is the most centralized place for Steam.