Where were you displaying this information to begin with? In the layout? So they'd see an empty select input with a description below it? That just feels like bad UX to display the empty select. The only way I can see to workaround this is under the Global tab of your query select add a custom blank value. Example as follows.

Value: (leave empty)
Label: - Select Event -

How could I add a custom HTML field below the query one and only display it during edition of the first query field and only if it's empty ? either by ajax or by profile edition ?

Create a Custom HTML field then add a Query condition for it to check if they'd have any options available so that the Custom HTML field only displays if there are no options. Next in the ajax settings of the query select field add your Custom HTML field to the "Additional Fields" parameter then set "Additional Field Titles" to "1 Line without Title".

You think it's better to have another custom HTML field or to leave it to CBIF in the query field description ?

Personally I think it's better UX to have it as a separate descriptive field, but it's entirely up to you. You'll just need to be sure the select can't be truly empty if you want it always visible if you still want to use your current usage.