It is natural to use "going to" in this context.
One sense of the "going to" future is to talk about future events that occur as a result or consequence of current conditions. So if you are asking "Do the current conditions of nausea indicate that you are likely to be sick in the future?", you would ask "are you going to be sick".
"Will" is a fairly neutral future tense, but it sometimes is used to suggest "future events that occur as a result of your choice". So "will you be sick" seems less likely, as it has shades of "is it your choice to be sick in the future".
Grammatically both are correct, and there isn't much difference. But I suspect most native speakers would instinctively choose "Are you going to be sick" more often that "Will you be sick", for this reason.