While I can't speak for the state of HI specifically, if the F4562 was not generated, then I would assume it's not needed. I left HI almost 20 years ago, and things have changed a bit in that time.
HI is mainly concerned with you paying taxes on your rental income, as well as the GET and if short term (180 days or less) they assess a TAT tax too. When it comes to taxes, the only state that taxes its residents more, is CA.
