I would like to submit my very current research to some Journal. The work is related to recent hurricane in USA (Hurricane Harvey). Since the peer-review process takes months to publish a work, so I am looking for some well-known journal which can publish my research in maximum 2 weeks with no cost. Would anyone have any idea where and how to publish very current research? or already published their very current research in past?
Quality peer review takes times: Science takes time and quality peer review takes time. Presumably, you want the stamp of approval that peer review brings. In which case, you will want to publish in a journal where the peer review has meaning. And in this case, peer review takes time. Sure, some fields and journals will get you an initial decision faster than others. But you might get rejected at the first outlet and that can take time; and you might be asked to do revisions somewhere and that takes time to get that outcome, and then for you to make the revisions, and then get an outcome on those revisions. The fastest experience that I've had from submission to in press was about 2 months. And I wouldn't expect to have the same outcome for other work submitted to the same journal.
Consider sharing a preprint: One option would be to post your work to a relevant preprint server. I don't know what discipline you are in, but OSF preprints will accept work in any discipline https://osf.io/preprints/
This would allow you to circulate and share your work immediately. You could then engage in the more time-consuming task of getting your work published in a peer-reviewed journal.
so I am looking for some well-known journal which can publish my research in maximum 2 weeks with no cost
This is unlikely to exist. There may be well known journals that will publish slowly at no cost, or obscure journals that will publish for free quickly, or predatory journals that will publish quickly for a cost.
But peer review takes time. So a proper publication.
What are you goals? If you want knowledge to be public, you could approach a magazine or newspaper, make a blog post, or post a preprint to ArXiV. This can also help if you want to establish work as yours, so you don't get "scooped".
At the end of the day, the urgent responses to a hurricane will likely come from government scientists, publishing government reports instead of peer-reviewed articles.