The chickens need protection from predators, and protection from bad weather (rain, snow, wind, etc). They should have a roost to sleep on at night, hens need a suitable place to lay their eggs, and of course they need food & water.
Some people provide those things with a coop, but it sounds like your covered run will probably work fine.
Since your run already keeps the quail safe from predators and bad weather, I'm guessing it will do those same things for chickens.
Because quail sleep on the ground but chickens like to roost up high, you may need to add a windbreak at the height of the roost. But you might already have trees or other buildings or something that keeps the wind off the roost. If you look at the chickens on the roost, and you can see that wind is ruffling their feathers, then you should add some amount of wind protection in cold weather. Ruffled feathers are fine in hot weather.
