So, is it possible to compile Compose Desktop only with IntellijIDEA and not with Android Studio?

Should be, as long as you have android SDK on your machine

Android SDK is not needed if we compile only-desktop project. Intellij IDEA can be used for opening desktop-only, android-only or MPP projects.

I think OP asked if you can compile compose-desktop itself, not a consumer project.

For compiling compose-desktop itself, yes, we need Android SDK. (Gradle build scripts refer to it even when we build desktop libraries)

I found the issue that didn’t show the Gradle’s tasks list On Android Studio, I had selected “Do not build Gradle task list during Gradle sync”

@Igor Demin but he means android studio, you can use the android sdk with intellij too. The only problem you can have is intellij doesn't support the latest AGP version

It's not a bug, it's a feature 😅