This is the image of an apple cut into half.
There are two apple halves, or two halves of an apple. (They mean exactly the same)
We can count the number of apple pieces.
A person might ask for half an apple.
This is a picture of some apple slices.
We can also say some sliced apples, if we know more than one apple has been sliced.
It's not easy to count all the slices, so it's just quicker to say some.
You could also ask for some apple if you wanted more than two slices or pieces.
This is a picture of a single apple and a quarter slice. As you can see, the apple is not cut into two slices.
The apple has not been cut into four quarters. (Alternatively) It has not been quartered.
Although we can see it is countable, some people might ask for some apple, or ask for the larger/smaller piece.
slice
a flat, often thin, piece of food that has been cut from a larger piece:
cucumber/lemon slices
Layer the pasta with slices of tomato.
Could you cut me a smaller slice?
quarter (noun)
one of four equal or almost equal parts of something; ¼:
He cut the orange into quarters.
quarter (verb)
[ Transitive; often passive ] to cut something into four parts
to divide something into four pieces of approximately the same size:
Peel and quarter the tomatoes and put them in the stew.
Cambridge Dictionary