hello mate. I would presume like any other situation you will be sent a non availability chit from your receiving unit which will give authority for private hiring. Please post again on monday when the guru's are back for a definitive answer.
(Been in your situation but quarter became available at last minute (thankfully in decent nick))
You shouldn't have a problem with getting an MQ in BFG. There are plenty.
Some garrisons have a large pool of hirings on long term rent, so even if they have to put yo uin one you will not really notice it. Apply for your quarter and wait out.
What makes you think you wont be getting a quarter? Or are you 'hoping'? Most quarters are pretty good now, they have been PUMA'd or STAR'd. The only real down side is you tend to be given the number of bed rooms you require and no more. WHere as in UK you can get one up in most locations.