There is no built-in way to share question over several sites, but the idea comes up with some frequency, see for example a feature-request to support asking on multiple sites.
While for general cross-site promotion there is the hot network question list (in the sidebar) and also the general http://stackexchange.com/ neither is designed to get attention for a particular question.
A workaround I am aware of is on the meta-site of MathOverflow where there is a question to collect questions on [math.SE] that users want to promote, this is however a bit of a special case.
Now, for your actual question: To get attention for a specific question from a specific site, I would join the chat of that site and mention the question there. Chat is quite informal, so this should be alright. If there is a lot going on in the room it might be good form to ask if it is alright before posting, if it is rather idle I would just post it in a polite way.
To post on the meta-site of that site is also a possibility; but this could be seen as slightly intrusive (especially if one should be a complete outsider to that site), yet it might still be an option. In any case, I would recommend to try chat first.