Is it possible to have access to the qid for the current question in the theme's question.twig file?
I tried but it does not seem to work.

No, because there may be several questions on a screen.

What are you trying to do?

I have a script in a custom Theme that tracks the user's mouse movement while he is answering a question.
Each question is displayed as a separate page.
I then store all this data in the Limesurvey database in custom tables.
If i had access to the QID i could store it alongside the movement data, making a query like "get all movements for question 2" very simple.

Try this:

{{ aQuestion.qid }} worked, thank you!

For anyone else that might have the same problem, an example code snippet: