Lorsque je saisi un ajout de nouvelle personne à partir d'une personne existante dont le nom comporte une apostrophe, "d'Irande" par exemple, le champ "Nom" ne se remplit pas automatiquement lorsque je saisi les informations dans les champs "Prénom(s)" et "Nom de famille".
It is not the apostrophe itself causing this behaviour.
By default the full name is constructed from given names + prefix + family name with spaces separating the parts.
From the screenshot it looks like you have entered the prefix and family name (maybe inherited from parent or induced from a child) and then edited the full name to remove the space following the apostrophe.
Once the full name is edited, webtrees stops auto-constricting it from the parts because it does not know how to. You have intervened, so it will stop doing that.
So next time, first fill in all the parts of the name, and wait until the end to edit the full name.
I have to apologise, after testing it seems this has a slightly different cause than what I initially thought.
Scenario: start with a person with full name eg John /d'Irlande/ and add a father.
I concluded the same behaviour as if the full name was edited appears when the family name is for example /d'Irlande/.
webtrees knows "d'" is a common prefix and extracts that. But since a space is missing, the reconstructed full name is different, so webtrees thinks it is edited.
You can prevent this by adding a space inbetween the prefix and family name, but that space should really not be there.