The first sentence reads to me as a run-on. It would be
Alex's son, John, is sleeping.
or
Alex's son John is sleeping.
John is an appositive, a noun or noun phrase that provides additional information or an alternative name for an adjacent noun. Whether or not you include the commas depends on what you are trying to express, as their inclusion or omission will change the reader's interpretation of the sentence.
If the commas are included, the reader will infer that John is Alex's only son, at least within the context of this passage or conversation. If that is the case, you would lose no essential information by omitting John altogether, or by swapping the noun in apposition, since they are equivalent:
Alex's son is sleeping.
John is sleeping.
Without the comma, on the other hand, the reader will assume that there is more than one son; John provides essential information for understanding the sentence by identifying which son is being discussed. Grammatically, it is called a restrictive appositive.
Alex's son John is sleeping [; however, his son Jacob is still running around].
If we remove John and Jacob from the above, we no longer know which son is engaged in which activity. And if we separate them with commas, readers would be confused because they will read the sentence as a single son having two names. If Alex has more than one son, therefore, you should omit the commas:
Alex's son John is sleeping.
