Looic on "get_terms : query by exact term_name, help needed"

Hello All!
I cannot making it work.

I have the following custom taxonomy structure (category like) :
A
\--My Sub Term Parent A
\--My Sub Term Parent Abis

Now I want to get the FIRST sub-term object so I query like that:

$query_term = array(
            'parent'        => 804, // id of term "A"
            'search'        => esc_attr("My Sub Term Parent A")
);
$cat_term_obj = get_terms($my_tax, $query_term);

The problem is it doesn't allow me to retrieve the exact match of the queried term name. It retrieves nothing (null). I tried with name__like parameter but it's the same.

Is there a way to retrieve a term in a custom taxonomy name, b ased on the exact term name ?

Thank you for your help,
Looic.