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It could be the Birmingham Boys Junior Football League (BBJFL) that's referred to. No team would have been called Birmingham Boys because of the representative nature of the name.
The Birmingham County Football Association in registering names of new football clubs take into account the area used in the name. A club would not be allowed to be named after the city because it could then be implied that that club represented the city, which it clearly would not.
I hope I've explained that well enough.
I checked the internet for the BBJFL but their (old?) site is unavailable which may suggest they no longer exist. You could ring Birmingham County FA and ask them for advice as every club has to register with them before they can join a league.