When both propellers of the aircraft reach the time limit (60 month) and need to be replaced, can both propellers be replaced at the same time and perform only one flight test to reduce time on ground or should they be replaced one at a time just for safety or to investigate any discrepancy that may arise?
Since replacing a propeller with the same propeller that is in the Type Certificate is not a major repair, the FAR(s) (CFR Title 14) that most apply are 43.7 and (assuming the airplane is under Part 91) 91.407.
(a) Each person performing maintenance, alteration, or preventive maintenance on an aircraft, engine, propeller, or appliance shall use the methods, techniques, and practices prescribed in the current manufacturer's maintenance manual or Instructions for Continued Airworthiness prepared by its manufacturer, or other methods, techniques, and practices acceptable to the Administrator, except as noted in §43.16. He shall use the tools, equipment, and test apparatus necessary to assure completion of the work in accordance with accepted industry practices. If special equipment or test apparatus is recommended by the manufacturer involved, he must use that equipment or apparatus or its equivalent acceptable to the Administrator.
(b) Each person maintaining or altering, or performing preventive maintenance, shall do that work in such a manner and use materials of such a quality, that the condition of the aircraft, airframe, aircraft engine, propeller, or appliance worked on will be at least equal to its original or properly altered condition (with regard to aerodynamic function, structural strength, resistance to vibration and deterioration, and other qualities affecting airworthiness).
So you must follow the maintenance manual and put the airplane back in a state at least as good as original. If you think that putting them both on does not qualify to satisfy this requirement, then you should do them one at a time. If it does satisfy this, then in that case, 91.407 comes into play, which states:
(a) No person may operate any aircraft that has undergone maintenance, preventive maintenance, rebuilding, or alteration unless—
(1) It has been approved for return to service by a person authorized under §43.7 of this chapter; and
(2) The maintenance record entry required by §43.9 or §43.11, as applicable, of this chapter has been made.
(b) No person may carry any person (other than crewmembers) in an aircraft that has been maintained, rebuilt, or altered in a manner that may have appreciably changed its flight characteristics or substantially affected its operation in flight until an appropriately rated pilot with at least a private pilot certificate flies the aircraft, makes an operational check of the maintenance performed or alteration made, and logs the flight in the aircraft records.
(c) The aircraft does not have to be flown as required by paragraph (b) of this section if, prior to flight, ground tests, inspection, or both show conclusively that the maintenance, preventive maintenance, rebuilding, or alteration has not appreciably changed the flight characteristics or substantially affected the flight operation of the aircraft.
I would argue that a propeller change cannot be satisfied by (c) above and does require a flight test. However, there is nothing that requires separate flight tests UNLESS the maintenance manual requires it, in which case part 43.7(a) from above comes into play. But as a "fail safe" a good run-up on the ground at high throttle with the brakes applied, followed by an inspection of the work can give good confidence in the work prior to the flight test.