The phrase "accidental quality" is correct and usable in written English.
It can be used to describe a characteristic or feature that is not intentional or planned, often in a context where the quality arises unexpectedly.
Example: "The accidental quality of the painting gave it a unique charm that captivated the viewers."
Alternatives: "unintentional characteristic" or "inadvertent trait".
The way you describe it, there's almost an accidental quality to Franks's career.
My Jewishness never appeared as anything else to me but an accidental quality, like, say, my blond mustache (which could have been brown).
"My Jewishness never appeared as anything else to me but an accidental quality, like, say my blond mustache," he writes to Zweig.
Khan often relied on old friends and university classmates as he constructed his supply network, and from the beginning there was a haphazard, almost accidental quality about his deals.
It is not a property of things nor an accidental quality, not a substance, and not a class of things.
Plato, beginning with the premise that sex is an accidental quality, concludes that women and men are equally qualified to become members of the guardian class.
(There are also accidental forms, which are a substance's accidental qualities).
In particular, he proposes, matter has essential and accidental qualities, though we can only know the accidental ones, through experience.
Because they too far away to make close observations, and too high in rank, we can only rely on inferences based on accidental qualities size, speed, and direction.
Consider how the usefulness and attractiveness of things produced by human beings are not accidental qualities when present but are a result of our intelligent nature and our natural delight in order, balance, and beauty.
In a similar way, it is unreasonable to conclude that the surpassing usefulness and attractiveness of things throughout nature are accidental qualities but it is rather more reasonable to conclude that they are the result of the existence of a surpassing intelligence which also delights in order, balance, and beauty.
