I coded the methodology based on the paper "Free Water Elimination and Mapping from Diffusion MRI" and I am curious if I am legally bound to take permission from the authors or the journal in making the code open source on Github?
You are not required to get permission to implement an idea you have read about in a paper, or to make it open source.
The paper is protected by copyright, but this only protects the text and images in the paper (the expression of the idea), not the idea itself. The copyright on the paper does not prevent you from creating your own realization of the idea in code and distributing it freely. You should also acknowledge the source of the idea by citing the paper in which you read about it, but this is an ethical requirement, not a legal one.
The exception to this is if the methodology in the paper is patented; in this case additional restrictions on what you can do will apply.