Quail are often quick to tame and can be handled easily. This does take some practice and a gentle yet firm hand is required to catch and keep hold of them. If you are holding your quail and you find it is desperate to go home then place it back in it's pen quickly rather then try to keep hold of it and risk either stressing it or losing it.
Quail should be caught by grabbing their body. You should never catch one by grabbing its legs. This will cause serious damage and could result in a broken leg.
With one hand under the quail, it is relatively easy to keep the quail quiet and do any health checks you need to. Handling your quail and getting it used to human contact will help you in the long-run even if you aren't keen to pick it up simply to stroke.
As you can see from the pictures there are two ways you can carry your quail easily, head forwards or head backwards.