This study investigates the multifractal formalism framework for quiet sleep detection in preterm babies. EEG recordings from 25 healthy preterm infants were used in order to evaluate the performance of multifractal measures for the detection of quiet sleep. Results indicate that multifractal analysis based on wavelet leaders is able to identify quiet sleep epochs, but the classifier performances seem to be highly affected by the infant's age. In particular, from the developed classifiers, the lowest area under the curve (AUC) has been obtained for EEG recordings at very young age (≤ 31 weeks post-menstrual age), and the maximum at full-term age (≥ 37 weeks post-menstrual age). The improvement in classification performances can be due to a change in the multifractality properties of neonatal EEG during the maturation of the infant, which makes the EEG sleep stages more distinguishable.