Benefits of quail egg; Quail egg is 3 times more nutritious than chicken egg and has 5 times higher iron content and high levels of amino acid. Amino acids are building blocks of proteins. In this sense quail egg is a good protein store for babies. It has high vitamin and mineral content. 
It is beneficial for development of babies and children, it increases appetite, heals tissues and boosts the immune system. It is good for asthma and allergy. 
It contains more iron and B12 than chicken egg. It is good for anemia.  It increases appetite as it contains high levels of Vitamin B1.  
Keep in mind 
As quail egg is very small, compared to normal eggs it is harder to crack it. Cut the center of the egg carefully with a knife and use this cut to crack the egg with your hand or knife. 
You should be conscious about giving quail eggs to babies. Even though quail eggs come with a lot benefits, its high protein content may put an excessive load on the kidneys of your babies that haven't grown fully yet. So it is better to limit quail eggs consumption for babies. Some researches emphasize that children who eat high amounts of quail eggs may go through puberty earlier than their peers... If you decide to give your baby quail eggs, please consult your doctor about the amount and frequency. And also definitely ask your doctor about the way these should be cooked.  
Nutritional value per 1 (one) (9gr.) quail egg;  
Vitamins Vitamin A- 48.9 IU Vitamin E - 0.1 mg Riboflavin - 0.1 mg Folate - 5.9 mcg Vitamin B12 - 0.1 mcg Pantothenic acid - 0.2 mg Chloine - 23.7 mg  
Minerals  Calcium- 5.8 mg Iron- 0.3 mg Magnesium - 1.2 mg Phosphor- 20.3 mg Potassium- 11.9 mg Sodium - 12.7 mg Zinc - 0.1 mg Selenium - 2.9 mcg