It doesn't sound idiomatic to me, for two reasons.
Firstly, a "shock" is normally something momentary that passes. "Culture shock" could, of course, be more prolonged than simply being startled by something, but still "having" it doesn't sound correct for the reason you mention - it sounds more permanent than it is likely to be. You will eventually get over culture shock.
Secondly, though, I suspect you mean "here" to refer to a specific place where you experience culture shock, such as a foreign country. However, the word "here" is also used informally to refer to a particular situation, for example, "I'm having some trouble here!" So, your example "I'm having culture shock here" could sound like you mean "I'm having culture shock now".
As mentioned, "shocks" are generally a momentary event, and it is more common to refer to that event as "a culture shock*". I think the most idiomatic say to say you are experiencing culture shock in a particular place would be:
This place is quite a culture shock for me.
The Merriam Webster dictionary gives some example usage of the term "culture shock" that may be helpful to you.