Could "I do not have any pillow" be an accurate translation for this sentence?
Absolutely, and it should be accepted now. :)
'ingen pute' can also be written out as 'ikke noen pute', which word for word translates to
'not any pillow'.
Um...by learning, ha ha. :P Dutch is actually pretty helpful, on top of Norwegian being pretty easy for me (and the adaptive lessons are shorter, so less time too).