There are no dumb questions from people who are eager to learn. Only to ask a question with no intention of learning is dumb.
It kind of depends on what You mean by green. If the tree is leafless, it is dormant. If it is nearly leafless (in the process of going dormant), and the wood is hardened (browning up in color), it should be fine. If it is truly green-colored wood with healthy green leaves, refrigeration will not induce dormancy, and the cutting will also probably not last long. Green cuttings are more tender and perishable.
There is a definition of dormancy, which applies more to grafting: you can take non-dormant wood, strip the leaves, and when you are done stripping the leaves, it is "dormant" - in the sense that it is leafless, and thus not actively using nutrients and water to sustain the leaves. The leaves are stripped when the wood is being used for grafting - but that is not what you are looking for in figs. It might be possible to strip the leaves from a branch, and leave it alone for a week or two (at this time of year) are to some extent force it into dormancy. That is just a guess.