子鹿って英語でなんて言うの？

Fawn is the term most often used to refer to a baby or young deer, and specifically refers to deer.
Calf is a term that refer to young ones in a certain species, and this term can also be applied to young deer.
Bambi was a very famous children's movie featuring a baby deer, so you could also playfully refer to a baby deer as a Bambi.

A young deer is usually know as a FAWN. However, when speaking about an animal such as a sheep, cow or foal that is a year old or that is in its second year, it may be referred to as a YEARLING. Young of deer or cattle or horses may also be referred to as CALFS.
'The Yearling' is a classic book written by Marjorie Kinman Rawlings about a child's relationship with a young deer.

Children deer of a stag (male deer) and hind (female deer) can be equally referred to by their zoological name or by a description. The use of either term depends on your audience. However, most English speakers will understand the word fawn. When used as an adjective, fawn also refers to a color (a yellowish tan)

The picture of a hind forsaking her newborn fawn is so contrary to her well-known solicitude for her offspring under normal conditions.

A young deer is typically called a fawn. The term 'kid' could also be used in relation to a baby deer, the same as it would be used in relation to a baby goat.

Using the word 'baby ~' also refers to a young animal.

The word 'doe' doesn't refer to a young deer but rather is refers to a female deer. I thought that adding this might be useful.

Example:
The doe gave birth to a beautiful brown fawn.