Sometimes yes, but not always, and it depends on the supplier the room came from. Some hotel suppliers allow a real amendment, where you send the new dates and the booking is updated while keeping the same reference. Others do not offer that at all, and the only way to change the stay is to cancel the original booking and make a new one for the longer dates.
In practice, cancel and rebook is the more common route. If the booking is still inside its free cancellation window, you can cancel it in the system and book the new dates straight away. The thing to watch is that the new booking is priced at whatever the room is going for on the day you rebook, so it may not match what your customer originally paid, and the room type they had may not be open for the extra night. Because of that, it is worth checking availability for the new dates before you cancel anything.
Where the room was sold as non refundable, or the cancellation deadline has already passed, a change usually means losing the original amount or paying a penalty set by the hotel. We cannot waive that, because the rule belongs to the supplier and the hotel, not to us. In those cases some agents book the extra night as a separate short booking and let the hotel link the two stays at the desk, which often works but is never guaranteed.
Before you promise anything to your customer, open the booking in your panel and look at the cancellation policy and deadline shown against it, then ask us which of the hotel suppliers on your account support direct amendments so you know which bookings can be changed cleanly.