Is there any ways to document Google search results in a research paper?
Is there any ways for me to show in my paper that this [Google] search provided these results at the time being?


No, this isn't possible: the same search can simultaneously produce different results, because Google considers factors beyond the supplied search term, e.g., location. 

That said, solutions are possible if you are willing to consider just the results of a single search. (Perhaps that's what you meant anyhow.) Leading to:

Yes (partly), you just need to record the raw result data (which will be authenticated, hence, cannot be forged*), which you can (probably) do using curl (with option --trace &lt;filename&gt;), for instance.

Ultimately, reproducibility is unattainable, but you can prove your results were derived by your method, as I have explained. (Your work might be useful without such proof.)



*There are some caveats.