Yes, birds do nap during the day when things get quiet. If he is eating, playing and otherwise active, he's probably just taking the little "cat" naps birds take.
If you see him sitting fluffed up a lot, or he goes off his feed or becomes inactive, then he is sick and will need to be taken to the Avian Vet for a diagnosis and treatment.
Keep an eye on him, but just taking a few naps during the day is normal and not a worry here. Other signs of illness to look for is a lot of sneezing, coughing, discharge from any body openings as these mean infection and must be aggressively treated. Once birds start going down, they go down hard and quickly, and it can take years to get them back to really healthy. This happens quite often with neglected birds who have been in a steady decline for a long time. Your bird sounds as if he is well taken care of. If you feel something is wrong, then by all means take him into see an Avian Vet as it is ALWAYS better to be safe than sorry.
