The TL;DR version:
"I am 20 years old next month" is incorrect because in that sentence "20 years old" is an adjective which describes you.
Remember that the verb 'to be' plus an adjective is used to talk about the present.
"It is my 20th birthday next month" is correct because the object of the sentence is the birthday, and is a scheduled event in the future.
Original answer follows:
I'm 20 years old next month.
Is understandable but grammatically incorrect, although frequently used colloquially.
As an aside. @Muzer comments that "It's not grammatically incorrect. -1.".
It is incorrect in that form, because if you want to use "I am" to speak about the future then it should be combined with a verb ending in "-ing". Here's how you can use "I am" correctly:
I am going to be 20 next month.
I'm turning 20 next month.
If you're talking about the birthday, as a scheduled event, then you can use the present tense like this:
It is my 20th birthday next month.
For more, see https://learnenglish.britishcouncil.org/english-grammar-reference/talking-about-the-future
I'll be 20 years old next month.
Is grammatically correct, but in my opinion:
I'll turn 20 next month.
Is the best way of saying what you want to say.
If you're currently 20 then it's not incorrect to say you'll be 20 next month, too; but to say that you will turn 20 suggests that your 20th birthday is next month.