Is there a way to calculate the number of all the languages over an alphabet?
It's uncountable infinite. 

There are the languages $\{ a^0 \}$,  $\{ a^1 \}$,  $\{ a^2 \}$,  $\{ a^3 \}$ etc., which is already countable infinite. 

But the powerset of the integers is uncountable infinite. So for every set S of integers, take the language $\{ a^k: k \in S \}$ and you have an uncountable infinite set. 

(The number of languages that can be defined by any algorithm is only countable infinite, so most of these uncountable many languages cannot be described in any way).