Obstructive sleep apnoea syndrome (OSAS) is the most severe condition on the spectrum of sleep-related breathing disorders (SRBDs). The Paediatric Sleep Questionnaire (PSQ) is one of the most used and validated screening tools, but it lacks the comprehensive assessment of some determinants of OSAS, specifically anamnestic assessment and sleep quality. This study aims to assess the accuracy of some specific items added to the original PSQ, particularly related to the patient’s anamnestic history and to the quality of sleep, for the screening of OSAS in a paediatric population living in Sicily (Italy). Fifteen specific items, divided into “anamnestic” and “related to sleep quality” were added to the original PSQ. The whole questionnaire was administered via a digital form to the parents of children at 4 schools (age range: 3–13 years). For each item, sensitivity and specificity, positive and negative predictive values, and positive and negative likelihood ratios were calculated. The highest sensitivity (80.0, 95% CI: 28.4; 99.5), in combination with the highest specificity (61.1, 95% CI: 35.7; 82.7), was found for the Item 32 (“assumption of bizarre or abnormal positions during sleep”). This item was found statistically significant for predicting the occurrence of OSAS in children (p-value ≤0.003). The study demonstrates the accuracy of specific items related to sleep quality disturbance for the preliminary assessment of the disease. Although these results should be validated on a larger sample of subjects, they suggest that including the factors discriminating sleep quality could further increase the efficiency and accuracy of PSQ.