do we add "''s" in the possesive case to the name that ends in "z"?
One of _____famous quotes is “;Si, se puede!” Yes, you can!
Chávezs’
Chávez’
(this is a task from my test and it says the correct answer is "Chavez'". I can't understand why)

Question

do we add "''s" in the possesive case to the name that ends in "z"?
One of  _____famous quotes is “;Si, se puede!” Yes, you can! 
Chávezs’
Chávez’
(this is a task from my test and it says the correct answer is "Chavez'". I can't understand why)

Answer

Actually, it is debatable. In singular possession, which is your case, some writers just put an apostrophe and others " 's ". In plural possession, it is certainly just an apostrophe.

Answer

Yes, the possesive for "Chavez" is "Chavez's". 
Here is a link to a page that I think will help.: https://www.bbc.co.uk/bitesize/topics/zvwwxnb/articles/zx9ydxs

Answer

Not really an expert, but I found this page
https://www.thepunctuationguide.com/apostrophe.html
which has an exception for nouns that seem plural, such as 
the United States’ lingering debt problem
.
Does not at first sight seem to be your case, but there is nothing else I see to explain it.
Is "Chávez" plural?