Hi, Guys. I believe that the way you do this is to cut the joist at the front and back end of where you want to install the stairs. Then put in doubled headers at the front and back of the rough opening. Then sister the joists on either side of the opening. But I'll bet that the stairs come with instructions to do this. For what it's worth, the rough opening will then be about 30-1/2" wide which is reasonable for the stairway. And since there is no load coming down from above like there would be if there was a live floor up in the attic, you should not have to build temporary supports.
But then again, the above should be checked and approved by someopne that knows a lot more about this than I do. I'll bet that there will be instructions with the stairs.