There is not a lot of research to date to answer this question. Some general recommendations can be made.
There are similar proteins in both hen and duck eggs. If you are allergic to hen eggs, your doctor will likely recommend that you avoid eating eggs from other domesticated animals. This includes duck, geese, quail, and turkey eggs. An allergic cross-reactivity can occur.
If your child really wants to try duck eggs, a prick skin test can be done in your allergist's office. The decision to perform an oral food challenge with this food could be considered as well. Discuss these issues with an allergy specialist.