Corals don't get these diseases, understand, but if you have a problem coral source, you may find that isolating everything without exposure to the creature on which they prey is a measure of safety. I have been at this for several decades without ever having needed that extra measure of precaution. You take your pick, take your chances. The lifespan of the velvet tomont is 3 days, but the tomite phase is indefinite---and it photosynthesizes, so you can't deprive it of food. Since ich and velvet both are sand dwellers during part of their life, and dependent on it, having no sand in a qt is a very good idea. Brooklynella is a ciliated animacule, which simply divides, so if you have these concerns I would recommend two sandless qt tanks, used in daily alternation, with filter cleaned and filter medium thrown out, aka tank transfer method. If you are still worried, 'wash' your specimen in a bowl of new saltwater before putting it in tank 2, etc.
You may take your choice of the precaution level required. A fish is the most dangerous thing you can put into your tank---but toward other fishes, not corals (unless it eats corals). It's a basic rule that things mostly travel on what they eat.