On my 13 QX56 (with Deluxe Tech), the spare was a 20", even though I had the 22's. I traded that for a 16 QX80 Limited and I just checked and that spare is also a 20". I'm pretty sure that if they told you that your QX came with a 22" spare along with the , they were just flat-out wrong.
Now, if they promised to replace the 20" with a 22", that's a different story and that's their nickel. But given that the Limited comes with everything Infiniti offer's on the QX80 and it doesn't have the 22" spare, I don't think that a 22" spare is an official option. You can check it out on the Infiniti website but I can't find where you can add a 22" spare.
2014 Q50 Hybrid Sport
2016 QX80 Limited