pourquoi on utilise 'règnent' ici, pas 'règne' ?

The phrase is “( Chez nous, Noël est une période ) où règnent le calme et la sérénité, “ 

This is stylistic inversion in the 2nd clause  - the subject of the verb régner being ‘ le calme et la sérénité ‘.

Hence the 3rd person plural form, which might be more obvious  when rewritten without inversion - ‘ ( Chez nous, Noël est une période ) où le calme et la sérénité (les deux) règnent’

 https://www.lawlessfrench.com/grammar/inversion-with-nouns/  

Bonjour coca,

I had some difficulty understanding exactly what was said.

The gentleman was referring to how they celebrate as a family — this is where I think I heard the verb régner, but in the third person plural present indicatif because he was making a reference to the family.

The third-person plural conjugation of the verb régner is règnent, which may be the answer to your query.

Bonne continuation 

Jim