This does sound like a bit of a pickle. If I read you correctly, you're emotionally bonded with both Q and Z (if they weren't together, you'd be the pivot in their V). I think RP might have underplayed your feelings for Z in crafting her response.
The first thing I'd suggest is simple: no ultimatums.
It sounds like things were simple and got complex. It is a natural response to try and simplify things again, and the fastest way to do that is with an ultimatum. It's also emotionally unfair. Sorting out feelings takes time, and the quick fix doesn't always work.
Next, I'd wonder about the origins of Q's discomfort.
I may be off the mark, but it could ne related to her self-image as a woman and her perception or her two men. In blunt terms, the idea.of her man with another man may cause her discomfort. She may feel her ability to satisfy is questionable if you need a man. She may see each if you as less manly for having feelings for another man.
I'd look into separating things out to see where the issue is. For example:
Does Q want you two only for her? (No other partners?)
What would Q think of you with another woman?
What would Q think of you with a man other than Z?
Of Z with another woman?
Of Z with a man other than you?
Other areas of concern would be the emotional and temporal questions of poly relationships:
Is it the physical activities that bother Q?
Or is it the time spent together?
Or the emotional closeness?
You made a good call by scheduling your Z time when Q was busy, but you're right in thinking you shouldn't have to always do that. I suspect though that some or the NRE that you have with Z may have made Q feel put out. In those cases, I find it's especially important to rearticulate from existing partners that they are valuable, how they are valuable, and that they will continue to be cherished.
I believe that if you have it can get answers to the questions above, you will be well on your way to finding a path. Just be aware that this is new ground for all involved, so sorting out those answers may be difficult or perhaps uncomfortable.
Best of luck.