Can we use a technique (equations and approach) from a journal paper in one section of our paper if we cite it like this: "Method X has been adopted from [Y], modified, and used in this paper"?
Method X has been adopted from [Y], modified, and used in this paper
I don't see why not. This is acceptable. In fact that is the most common form of citations you tend to see in papers that make an improvement over an existing method.
Just make sure you know ample knowledge of what you cite in this case as you are to make a modification to what is published. It would be best to know well about what X really is before you try to modify it.