While reading the paper of some authors. I have decided to work on a question which they listed as an open question in the conclusion section. I'm hesitating on whether or not to cite the paper, from which I got the question, because usually we cite papers providing answers rather than papers stating questions.
Is is unethical to work on the question on my own paper without giving proper reference to the paper I got the research question from? if so, how should I cite the paper?
Yes, I think you should. Questions are at least as difficult to develop as answers, I would not privilege one over the other.
You need not cite it as explicitly as "We got our research question from (Andy, et al. 2022)", though.
Rather, typically in the introduction of a paper you want to set up what current knowledge in the field is and what issue the paper at hand is addressing and why that is important. One of the most straightforward ways to make a case that the problem you are addressing is important is to say that someone else also thought the problem was important! Or at least, important enough to mention in their own paper as an open question. You can cite this motivating paper as you make your case.
Yes, you should absolutely cite the paper. In addition to the reasons explained in Bryan Krause's anwer, here is another important reason:
People who read the open question in the paper might want to know whether the question has been solved meanwhile. The first thing they'll probably do to find out is to check where the paper was cited. So by citing you will help those who are (or would be) interested in your work to actually find your work.
By the way, I really don't see any reason at all not to cite the paper. There's simply no disadvantage in citing it.
Usually it is difficult to proof that your research is relevant. You get a witness for free!
Further, it gives credit to the author which is justified.
Citing a paper that inspired your research is extremely common; refer to it the way you would any other background information that situates your work.
Some examples from a quick search of the arXiv (quotes are taken from the final published versions):
In this realm we may classify forcing notions using the methods of [23], [24] and, for example, declare that very Souslin (or generally ω–nw–nep) ccc forcing notions (see 1.3.1) are really nice. Both the Cohen forcing notion and the random forcing notion and their FS iterations (and nice subforcings) are all ccc ω—nw—nep, and [22, Problem 4.24] asked if we have more examples. It occurs that our method relatively easily results in very Souslin ccc forcing notions (see 1.3.4(3), 1.5.8(2), 1.5.11, 1.5.15(3))
…
[22] Shelah, S.: On what I do not understand (and have something to say). Fundamenta Math. 166, 1—82 (2000) math.LO/9906113
In particular, we answer the question of Erdős et al. [8, Problem B], who asked for the asymptotics of r̂(Ks,n, Ks,n).
…
[8] P. Erdős, R. J. Faudree, C. C. Rousseau, and R. H. Schelp, The size Ramsey number, Period. Math. Hungar., 9 (1978), pp. 145–161.
I believe that you should cite the paper. One fundamental reason is that context is hugely important in conducting research and framing it properly for the purpose of publication. Here you seem to have gained a lot of the perspective you need in order to ask the main question for your paper from the paper you are referring to. As such, I think it warrants a citation.
The other answers have pointed out that the answer is yes, you should absolutely cite the paper you got your questions from.
I just want to emphasize how narrow a view
because usually we cite papers providing answers rather than papers stating questions.
is.
Providing answers is a reason to cite a paper, but far from the only reason. Citations are not a precious resource only to be used when absolutely necessary. It costs you nothing, and helps your reader, to cite papers “merely” because they provide relevant context for your result.
