Is there any way to make a browser recording in IntelliJ IDEA, how it is possible in Katalon Studio or Selenium IDE?
Katalon Studio is a test automation product that includes a web recorder and an Eclipse-based code editor (as well as some other stuff like API testing tool, object repository, etc.).
IntelliJ IDEA is an integrated development environment.
So they don't have the same purpose as Katalon Studio is a software that combines many functionalities, one of them is a part of an integrated development environment.
One thing you can do is to use Selenium IDE to record a script and then export it to one of the following programming language / testing libraries:
So you can export it to a Java/JUnit script and open it in IntelliJ.