Hello and welcome to the forums,
Yes, you should move the chicks. Ensure the nest box has a concave bottom lined with clean wood chips, not cotton or paper. Be sure to do so gently and move quickly while the mother comes out of the coconut for water or food. She will go in to feed them into the new nestbox. Try and ensure that when you touch the chicks, your hands are warm!
Is the father in with the chicks at this time?
This is a less than ideal situation as the lack of nestbox indicates that it was an unplanned breeding, correct?
I am putting the below as a note to ALL
members who may see this post, as we are primarily an educational forum:
❗Please be sure never to put any type of coconut hut or nest box in the cage. Budgies are much healthier and happier when they are never bred, and it is essential to prevent breeding when you have mixed genders. Breeding, especially when unprepared or uneducated on the essentials, can be seriously detrimental to the parents and chicks, and many situations can arise such as aggression between parents or parents and chicks, potential chick abandonment, malnutrition, aggression towards the owner, plucking, etc. etc. People who breed budgies must not only know how to respond in any of these situations but also have an established connection with a reliable avian vet, full knowledge of a budgie breeding diet, how to handle chicks (and feed chicks in the case of an emergency), understand what complications can look like (such as egg binding, etc) to ensure ASAP transfer to an emergency care facility, etc. For more information, please read the links below:
When We Don't Want Eggs!
A Breeder's Tips for Discouraging Breeding
Before You Ever Consider Breeding Your Budgies
Guidance for Breeding Advice Threads
Now, onto your specific situation: there's nothing that you can do about having chicks now so we will try to help you save the chicks as best as we can. The first thing I would do is ensure you have a good avian vet; you already should since you own budgies but the reality is that many people live in places without access to an avian vet. If you do not already have contact with one, please try and phone all exotic vets in your area and explain the situation. It's best to set up these relationships early so that if there is a problem with the chicks you can take them there and they are aware of their background.
Moving the chicks is the first step. Other staff members with more knowledge in this area will respond shortly
Do you have photos you can share of your current cage setup? What is their diet like? Does the female have a cuttlebone in the cage? How old are the parents?
Apologies for all the questions, but it's essential you provide answers to all of them so that we can better understand the situation and give these chicks a fighting chance
Meanwhile, please be sure to read through the other forum links in your spare time. You'll find everything you need to know to stay posted on the best of budgie care practices. If you have any questions after reading through things, be sure to ask as we'd love to help.
We hope to hear updates soon!