It would be:

Joséphine a les cheveux bruns et blancs, mais Simon a les cheveux gris.

We use the definite article in this case.

In particular French definite article is frequently present with names of parts of the body:

Elle ferme les yeux. She closes her eyes.

Ouvre la bouche ! Open your mouth!

Note that English has the possessives (my, his, her and the like).

See for instance here: https://french.kwiziq.com/revision/grammar/using-le-la-les-with-body-parts-and-clothing-definite-articles

See also the classic A Comprehensive French Grammar

Complément de réponse

As stated in user Dimitris‘ answer the normal use in French is a definite article, whereas in English a possessive is always used.

There is no difficulty in knowing which is correct from the definite article and the partitive:

• English: possessive ===> French: definite article
• English: partitive ===> French: partitive

You only have to ask yourself the question every time: is it the whole thing or just part of it? It soon becomes a triviality to decide which to use.

• John is fair haired, but his hair is not as fair as when he was young.
  
  John a les cheveux blonds, mais plus aussi blonds que lorsqu’il était jeune.

• Mister Jean is not old but, nevertheless, he has some gray hair.
  Monsieur Jean n’est pas vieux mais il a quand même des cheveux gris.

• Terry pulled his sister’s hair.
  Terry a tiré les cheveux de sa seur.

• Terry pulled some hair off his sister’s head by pulling violently on it.
  Terry a arraché des cheveux de la tête de sa seur en tirant dessus violemment.

There are other possibilities when the idea of a partitive is hard to conceive.

• She has hair on her legs.
  Elle a du poil aux jambes.

In English, the idea of a partitive is not always made explicit: in the above sentence there is no word expressing, in itself, the notion of a partitive.
However, you can also say “She has some hair on her legs” and as you say so ask yourself whether you say something much different or not; you’re bound to tell yourself it’s quite the same thing. In French there is no choice: whether mass nouns or countable nouns, an explicit partitive has to be used; notice that when you say “Her hair is plaited.” you are using the countable noun; when saying “She pulled (some) hair off my head.” you are using the mass noun (uncountable). The hair on the legs is considered as nothing else but as a part of a mass, that is most of the time; however, not so rarely, you say instead of “She has hair on her legs”, “She has hairs on her legs.”. The same is true in French, as “poil” is both countable and uncountable. The difference is that in French an explicit partitive has to be used: “du poil” (necessary contraction of “de le poil”) for the uncountable noun and “des poils” (necessary contraction for “de les poils”).

In French you use rather the uncountable word.
(another example and yet another)

The question one might have is “How come that when you’re talking about only a part of the hair on the head you use a partitive, which seems correct, but when you talk about all the hair on your legs you also use a partitive?” (Notice that there is no difference in either language, that’s been made clear in the above). The answer is that we use two different references for a whole, an unexplicitly acknowledged convention: all the hair on the head on the one hand and all the hair in a rather not well defined whole of human bodily hair (not in creation, but in modern times perhaps). Another question is “why do that,”; it can be supposed that this has been conceived so, more or less unconsciously perhaps, but if we think about it, judiciously, because the hair on the head is very exceptionnaly absent, very important, whereas bodily hair is not nessarily found in all cases (even men can have rather little bodily hair whereas they always have hair on their skull).