Not gonna happen since you need two weeks off to participate in the Indy practice + qualifying.
You can have another driver qualifying the car. Not very common, I think the last time that happened was in 2015 when Tristan Vautier qualified in place of James Davison who was racing in Pirelli World Challenge.
It’s surely not an ideal way to make an attempt at the 500 but doable when F1 is racing in North America. I mean, you can do the practice days Tuesday to Friday, fly to Montreal for the weekend, and fly back for more Indy testing on Monday.
Of course, you need a solid substitute driver to qualify the car and not get bumped if there are more than 33 entrants.