It should be possible to import bitmap images (like the PNG image) to Adobe Illustrator and work with them in your project, it's not a conversion in true sense, just image import.
However, you may have been looking for a way to create a vector AI graphics from PNG image. This will require moderate knowledge of Illustrator and its functions, especially one called LiveTrace. This function basically allows bitmap to vector conversion, but requires that the person doing this tracing knows what he is doing.
You can find more about LiveTrace function in this excellent tutorial Converting Rasters to Vectors Using Live Trace in Illustrator.