At sufficiently low temperatures, quantum degeneracy of gas particles becomes important and an ideal pas deviates from the classical ideal-gas behaviour. In such a case, an ideal gas is called a quantum ideal gas. For quantum ideal gases, a corrected equation of state, which considers the quantum behaviour of gas particles. is used instead of the classical one. It is valid for both quantum and classical ideal-gases and it is reduced to a classical ideal-gas equation-of-state, under the classical gas conditions. There are two types of quantum ideal-gases. One of them is the Bose type and the other is the Fermi type. Here, Brayton refrigeration cycles working with Bose and Fermi type ideal quantum gases are considered and they are called Bose and Fermi Brayton cycles respectively. Coefficients of performance and refrigeration loads of these cycles are derived by using the corrected equation of state. It is seen that refrigeration loads are different from those of the classical Brayton cycle, which works with the classical ideal gas. On the other hand, coefficients of performance of these cycles are not effected by the quantum degeneracy of the refrigerant and they are the same as that of the classical cycle. Variations of the refrigeration load with low temperature (T-L) and low pressure (P-L) are examined. Under the quantum degeneracy conditions, it is shown that the refrigeration load of the Bose Brayton cycle is always greater than that of the classical Brayton cycle. On the contrary, the refrigeration load of the Fermi Brayton cycle is always lower than that of the classical one. Moreover, the minimum value of T-L for the Bose Brayton cycle is restricted by the Bose-Einstein condensation temperature for a given value of P-L.