Earlier this year I asked a question about Joe Wilson Junior actor. 
When I search for him on the net my question always pops up.  I do NOT
 want anyone to know about my query.  Do they?  or does this just come
up on my computer?

Yes...everyone can see it...
As per the Google Answers Terms of Service
http://answers.google.com/answers/termsofservice.html

Confidentiality; Use of Information. The questions or comments you
submit and the responses you receive do not form the basis for a
professional-client relationship, including but not limited to
attorney-client, physician-patient, or therapist-patient and are not
subject to standards or requirements of confidentiality and/or
privilege associated therewith. Furthermore, Google shall have the
perpetual, irrevocable, non-exclusive right to use, reproduce, modify,
edit, translate, publish, perform, display, post, transmit and
distribute ("Use") your questions, comments, and/or the corresponding
answers without compensation to you, anonymously or in the aggregate,
for internal or external purposes, alone or as part of other works in
any form, media, or technology, whether now known or hereafter
developed and to sublicense such rights. Google will not, however,
publicly disclose your questions or comments in a manner that directly
associates them with you (except where you include personally
identifiable information in a question or comment and such question or
comment is Used by Google as provided).

Hopefully you chose an ID that isn't personally identifiable (such as
mine "canadianhelper" doesn't tell you anything about me necessarily
and is a name I use NO WHERE else other than on this site).

I don't understand at all your answer to the question as to whether
anyone on the internet can see that I have asked a question about
someone--other than me--the one who asked the question.  You know,
sometimes you want no one to know you've aked the question!