One of the pretty obvious solutions is to actively promote Data Science SE site by mentioning it on relevant web sites, blogs, etc. Specifically, on Quora, you can mention this site in your answers as well as linking to relevant questions or answers here.
Having said that, while DS SE is a dedicated data science site, unfortunately I have to disagree that it is more professional (which should be the case, in theory), as I see a large number of high quality questions, answers and posts on Quora in relevant categories.
Quora has its own share of problems, discussing which is beyond the scope of this answer (and I don't have time for that, anyway), however, many experienced data scientists are active on Quora and likely don't see reasons to split their activities between it and DS SE. It is a problem for this site and I don't have a perfect solution for that.
As for "flood of doubts and articles ... about data science", promoting this site IMHO will not change that at all. This is because all those "doubts and articles" are less of factual nature and more of opinions and discussions nature, which, by design, clearly represent a better fit for sites, such as Quora, forums, blogs, etc., rather than more factual-based sites, such as Data Science SE.