Nope, that's not how it works, sorry. They may send the first check or two paper but then they stop. Knowing how payments for disabled veterans are issued is my job, plus I'm a disabled vet myself. 2013 is the year the VA put a stop to issuing paper checks beyond the first one (sometimes very rarely the second one, but then they WILL stop sending until dd is set up).
Certainly someone like Quinn would not be receiving paper checks even if your thesis were true or if we accept that the real world doesn't inform the show (which, to be fair, it often doesn't). There's simply no way he would not have direct deposits already set up. Not only is he a prior government employee, not only does he have a caretaker in Carrie who clearly would have done this for him seeing as she's the one who apparently helped him enact his disability pay as well as care at the VA, but the VA itself would have made sure to perform this function the moment he started long term residential treatment at one of their facilities.
Of course, it's also completely unbelievable that he received disability benefits in such a short period of time, but I find that easier to handwave since he was CIA and someone probably knew someone who knew someone who could push it through mega fast.