I own quails, finches and budgies, and house them all in the same aviary. I have never had any issues with any fighting, and my budgies don't even glance in the quail's direction. The quails have successfully raised three clutches with my budgies present, and none of the chicks were harmed. However, when I was selling the baby quails, a customer and I got into a discussion about bird compatibility.
He told me that he was amazed that I was able to house finches, quails and budgies together, as he had a sad experience a few years back.
He himself had an aviary with two pairs of budgies, a pair of cockatiels and a trio of quails. All was well until the quails reached maturity and hatched a clutch. He told me that one of the budgies went on a rampage, sadly killing all the chicks and the adult quails,even though they were completely fine with each other before the baby quails hatched. He was only purchasing more a few years later, as his last budgie had unfortunately passed away recently, and he wished to get back into quails.
So although my experiences have been happy, I would advise you to be very careful. I think it's really depends on the individual bird. Mine are calm and laid back, and are not territorial or aggressive. You may be inclined to house the budgies separately for a while, or maybe just place them in a cage in the aviary so they get used to the chicks. Also, if it is impossible for you to seperate them, it would be a good idea to provide lots of shelter and cover for the quails in the form of plants, logs and rocks.
Hope this helps and good luck !