How much space does the mini coop/run have? Do you have a photo or link to the product you're considering?
My first instinct is - not a good idea to combine the Quail with Chickens. There are some diseases that more readily affect Quail.
If it's a coop with a ramp the Quail will likely not go inside housing, they will sleep on the ground or in a box on the ground - is the run covered/roofed? Is there plenty of room for you to place some hides for the Quail? Is the run very secure against rats/mice and predators?
I get that you are looking for solutions, but it may be better to have a hutch or cage for the Quail and a separate coop/run for the chickens.
I also understand finances/budget may be a factor.
TSC has a XXL dog cage that is 48.5"L x 30.25"W x 32.5"H this can be covered with hardware cloth and would be a suitable size for 4 Quail. It would still need something to cover it to keep out weather/wind, like a tarp or similar. And getting it up off the ground would be a good idea too.
Honestly, some of the prefab products will not last very long out in the elements. If you can provide cover/shelter for them, they will last longer. But as a short term solution, sometimes it's better than nothing.
You will also want to consider how much space is for the 2 hens (chickens) as well. They still need some room to be comfy.
So, if possible, house them separately.
Just my thoughts.