"Her question is serious."
is there a reason I cannot write: dia pertanyaan serius?
The word order is incorrect.
The head noun comes first and is followed by the modifiers.
"Pertanyaan" is the head noun and comes first.
"dia" functions as a possessive (her question/his question) and comes after the head noun.
"Pertanyaan dia serius." = His/Her question is serious.
"Pertayaannya serius." = His/Her question is serious.
Regarding your word order, 'dia pertanyaan serius' :
In this word order, 'pertanyaan' is modified by 'serius'.
'pertanyaan' is the head noun.
'serius' modifies the head noun.
'pertanyaan serius' = 'serious question.'
'dia pertanyaan serius' = 'He/She is a serious question.'
You can use your word order like this (in a slightly different sentence):
"Dia punya pertanyaan serius" ==> He/She has a serious question.