I know this one already has an accepted answer, but I think the flight mechanics part deserves a bit more science.
Flight is all going to come down to two factors: the wyvern's profile and the square-cube law. We do not know very much about quetzalcoatlu; so, the best place to start is to look at the most powerful pound-per-pound flying animal we have alive today, and see if we can make it work. For this I will be using the crowned eagle which is an 11kg bird able to carry prey up to 20kg back to its nest.
A mid-sized quetzalcoatlus is about 20 times the mass of a crowned eagle. Thanks to the square cube law, we know that it is unreasonable to expect it to perform to the same ratios as the eagle, because mass cubed can only carry a squared capacity.
Why quetzalcoatlu size does not work
- The combined mass of a crowned eagle carrying large prey is about 31kg
- A mid-sized quetzalcoatlus weighs about 220kg.
- The maximum lift capacity of a quetzalcoatlus sized eagle like animal would be ((20^-3)^2)*31 = ~228kg
This means that even mimicking the most powerful pound per pound birds we have today, your wyvern would be just barely able to fly for fairly short distances under its own strength, and could absolutely not carry a rider at all.
Is there a more optimal size?
After running a number of calculations, I found that the ideal size for a bird to maximize its total carry capacity is about 75-80 kg with a total carry capacity of ~37kg. 37kg is pretty darn impressive, but not enough to carry almost any adult humans. However, children in the 8-12 year old range (or adults with significant dwarfism) would be small enough to ride an 80kg wyvern for short distances.
Would a more optimal shape help?
Perhaps... larger animals almost always have adaptations to help support their greater size that smaller animals do not. Lower bone density, less mobile joints, more complex circulatory systems, etc. The square-cube law assumes that you are just taking the eagle and making it proportionally bigger in all three directions. The Eagle is an aggressive predator meaning it has a certain amount of brawniness it could do without if you want an animal that focus on pure flying efficiency. If instead of making it ~1.91 times the size of the eagle in all directions, what it you made it ~1.5 times as thick (or similar proportions, just 30% less dense) and ~2.16 times as wide and long. Then you would have an animal ~77kg with ~4.67 times the lift area of the eagle giving you ~68kg of lifting power which is the approximate weight of an average pre-industrial grown man.
The takeaway here is that making a rideable avion like creature may just barely be doable as long as you build it to pretty specific specifications. Ideally, it should have a wingspan of ~25.5 feet, and a length of ~13.5 feet with very broad wings like an eagle, owl, or flying fox and weigh in at ~77kg. Its peripherals would all need to be pretty minimal in size too; so, I would shy away from horns or a long reptile like neck or tail. Instead, you might want to give it horn and tail like feathers (which some birds have). This way you can get more of the look you want without compromising the carry weight.
As for fire breath... there is no example of this is nature, but... dragon breath is so mainstream in our fiction that most people will not question it.
