

As per the 1099, the Royalties were over $1,000. Is this not considered income? At any rate I selected the Top No and the Bottom Yes to see what would happen and the X is still in the Not Required to file. This is more a curiosity question but probably doesn't impact why the TurboTax fields are wrong.
I went to the article you referenced and it looks like deversd was supposed to be there best answer. I can get to the Oklahoma button below the Personal Info but nothing else matches what they describe after that.
To get to that screen. Open TurboTax, click on the STATE TAXES tab on the top (to the right of the FEDERAL TAXES tab). It should say “Let’s work on your state return.” Click on the Oklahoma button which is right below the “PERSONAL INFO” tab. “Federal Agency Bond Interest” question appears. I didn’t have this so I clicked no. The next screen is “Adjustments to Federal Income” click Done. The next page is “Summary of Oklahoma Capital Gains and Losses.” Then you click “Edit” for the gain/loss shown
Regarding 561-NR, here are the property types per the instructions:
Enter the number in the box which corresponds to the type of property sold:
Again, not sure why the options are only for OK since the OK Capital gains should all be zero. Turbotax did set the values for OK to zero so it should be OK but I was hoping you might recommend a good value for A3. My guess would be 1 because it was stocks, even though they were not OK stocks.
Here is what I got back from Oklahoma.

The reason I blacked out my personal information on the turbotax forms was so you could see what Turbotax filled out. In summary,
511-NR
1. OK zero
2. both set to adjusted income
13. zero
17. zero
18. a blank, b blank % blank
19. zero
25. both $152
31. $152 not listed on OK adjusted
32. zero not listed on OK adjusted
33. $152 not listed on OK adjusted
34. $152 not listed on OK adjusted
38. $152 $30 on OK adjusted
Hope this helps.
Your issue doesn't seem to be capital gains. It appears that the difference is the $3,036. Oklahoma is adjusting that as taxable, whereas your return had zero listed on Line 1 (Oklahoma source income). Since the royalties are taxable to the state, the adjustment made by the state is correct.
You are required to allocate the Oklahoma sourced income items to the state on the screens, Oklahoma Income. If no number is entered under the Oklahoma amount, then nothing will be on the return as taxable.
Go back through the screens and allocate your royalty income to Oklahoma. An example of this screen is attached below. In this example, I entered a royalty of $5,000 and there was a depletion deduction taken. The total nets out to $4,250. This amount flowed to the Oklahoma nonresident return.

@jschimmer