This is an issue across the SE network. Many people will ask in chat instead of the site. This is annoying for various reasons:
When asking in chat, nobody else benefits from the answer. The site doesn't gain any new content and someone with the same question in the future won't find the solution here.
Asking a question on the site tends to force people to formulate it clearly and give the necessary background information for it to be answered. Getting this information in chat is often painful, time-consuming and annoying.
So, I tend to ask people to post a question on the site and then come to chat to discuss it, if necessary. That seems like the best of both worlds.
Now, there is no way of bringing questions from chat to the site short of someone manually reposting them as questions. Therefore, "migrating" is not an option, sadly. All we can do is ask the user to post a question or, if we find it particularly interesting, to post the question ourselves.