No. Don't copy-paste the same question onto another site. Instead, if the original question was never answered, do one of the two following things:
If the question is less than 60 days old, suggest to the original poster that they consider migrating it (they can request it be migrated by flagging it for moderator attention and asking the moderator to migrate it). The benefit of migration is then we don't have two copies of the same question left open (people will be automatically directed to answer on a single site), and there will be bidirectional links both ways. But it's up to the OP -- respect their decision, and don't pressure them.
Alternatively, when you post on the new site, ask a new question that is tailored to the audience of the new site. Yes, the original might be technically on-topic for the new site as is, but there are almost always ways you can improve the question and make it better tailored for the new site. Don't be lazy.
Also, link back to the previous version, and leave a comment on the original linking to the new version, so everyone can find both copies. And don't just copy blindly without any further thought: spend some time searching and trying to solve the problem yourself before posting it on the new site, and show what you tried in the new question.
The above analysis assumes the original was never answered.
If the original question already received an answer, don't copy it onto a new site! The world doesn't need another clone of the information -- that's not advancing our mission. A significant part of our mission is to build up an archive of high-quality questions and answers; but if the question has already been answered, then posting it on a new site doesn't serve that purpose, so you should not re-post.
(Just to be clear: per your statement in the question, this answer is based on the assumption that you are not the person who wrote the original question.)