I think you are asking about the inversion here : “complimentaient des lèvres rouges … ” rather than “que des lèvres rouges …. complimentaient” ? This is a formal and literary style choice, commonly used in writing.  (See Stylistic choice - after relative pronouns ) 

https://www.lawlessfrench.com/grammar/inversion-with-nouns/

Hi Liyunzi,

"Son nom lui venait de sa peau diaphane 'que' complimentaient des lèvres rouges comme le sang."

Conjunction "que" is "that" in English referring to "peau" the object. "qui" would not be appropriate in this case which would refer to the subject (Son nom), not the object "peau".

Hope this helps.

Bonne continuation

Jim