The question is asking for ideas to attract (potentially) knowledge contributors, not whether Quora's format, policies and etiquette are compatible or comparable with LawSE.1
I'm not familiar with Quora, so generally speaking, social media is probably one of the better ways to promote anything and everything these days - Twitter and Facebook seem to be the front runners but there may be others out there.
I assume, without any basis in fact whatsoever, that there would little or no enthusiasm from Stack Overflow or its subsidiaries to support / finance any sort of marketing campaign. Therefore it would be up to individual users to put in the effort by, for example, sharing links to questions and / or answers in a way that would initiate a conversation and attract contributors - if they don't "fall in line" with our policies and procedures then so be it, but if they do then it's job done.
Also, as well as playing the percentages game by sharing multiple links in this way, there's the added bonus of the Announcer, Booster and even the rarest of rare Publicist badges.
1I suspect there are many LawSE users who use different composition styles depending on the particular audience, site, or forum; some may even be active on Quora.