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 <filename>), 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.
I think copy and pasting the text or making screenshots is enough.
Google search results are out of your control, therefore not deterministic from your perspective.
If you want an additional integrity or certification, I think you are asking for too much. No research that I can think of comes with this kind of attestation. A biological research paper does not have an appendix with a sworn and signed statement saying: "I, cell 4588, hereby solemny swear that I divided 20% faster after being injected with chemical X."
Strictly speaking, nobody believes you until your results have been reproduced by other parties, anyway. Until then your paper is just a claim.
Seeing how Google search results are out of your control, the best you can do is describe what observations and deductions you made and then theorize on the hidden (deterministic?) model behind them. Screenshots or text results would then merely serve to illustrate your observations.
This is why having a detailed methodology section is important. The more details you provide, the better other scientist can approximate the circumstances of your experiments. Details could be your location, time, whether you cleared your cookies, whether you were signed in, operating system, etc.