Alright, well, total honesty? I don't give a crap about your little ship there. But I'll tell you what I do give a crap about, medieval and ancient law codes.
So, the theory of having a commoner become a queen sounds like some crappy self-insert fanfic of an idea, but it actually happened several times. Hell, the great Suleiman the Magnificent married a slave prostitute, Emperor Justinian married Theodora who's interesting life before becoming empress involved being a wool spinner, a dancer, a prostitute, an actress, and a paid live-in woman for at least 2 other aristocrats. And let me tell you Theodora was brilliant, she maneuvered the political system as adeptly if not better than her husband.
Then if you want to get into what we think of as pure medieval cultures you'll find at the tail end of the middle ages Edward IV who married Elizabeth Woodville, the daughter of a landed knight. But you really wont find all that many. And here's why: the marriage of a king was an immensely political affair, it had to strengthen the power of the crown in some way, and a commoner doesn't do that. The general rules of thumb: if the ruler is strong enough by himself that he doesn't need to have his marriage be political he could marry anyone he likes. It's still not advised, but he could. But, for the vast majority of medieval history other political families can always be nearly as powerful as the ruler and would therefore need to be controlled.
But that's with kingdoms. Demacia (at least originally) is a City-State. There we have different rules apply. For instance, since city-states are a lot smaller, it is far more likely that the central authority, be they king or not, has less to fear of giant political rivals. In fact, it's unlikely for city-states to even have a rigorously strict separation between noble and commoner and ruler. Not that these distinctions didn't exist, but, there tended to be some leniency of people rising or lowering themselves up and down the political hierarchy.
With exceptions of course, slaves in many city-states remained slaves, Sparta was known for basically only allowing downward mobility and not upward. But at the same time there were only 3 steps in their social hierarchy to begin with.
And then we just get to the weird as hell exceptions that make no sense but they just happened. History shows us anything is possible, like a Joan of Arc figure, who was the daughter of a tax collector and farmer, yet somehow became the commander of a medieval army. It makes no sense, but strange things happen.
As to whether or not Quinn could thrive in a political environment, that would largely depend on her temperament and how cutthroat Demacian court is. I somehow doubt that they'll straight kill her, but if Demacia does have powerful families that need to be finessed into working with the throne and if she is incapable of showing proper etiquette and making deals then she will be a failure as a queen. Assuming, of course, J4 doesn't handle all of that directly himself. In which case the worst Quinn could do is say something wrong to the wrong fellow and be required to make a public show of affection and apology. From what little I know about Quinn, I doubt she would be a Theodora. She might be a passable somewhat uncouth queen that would pass on to obscurity in the next few generations. With maybe commoners remembering her as some folk hero who raised herself to nobility, and nobles remembering her for being somewhat incompetent and a reminder why they don't usually do that.
Unless, she did something truly amazing to get herself remembered. If she leads an army and wins some great victory. Or organizes the best spy ring in Demacia. Or, uses her scouting skills to actively hunt down J4's political opponents within the city. Then she'll be remembered as either good or evil, depending on her methods and her successes.
