The 36" or so LP hose with quick-connects that came with the trailer is made by Fairview
(per the tag on the hose.)
After some Googling around, I found that the female quick-connect housing and the male nipple are from the Gas-Flo RV series in the online Fairview catalog. They are the 1/4" size.
I'll be simply getting an extra full-flow nipple (the Fairview P/N QD-GMRVN4-4M) which I'll adapt to the Weber Baby Q. The male nipple has 1/4" MPT threads and the Weber probably has 1/8 MPT threads (not sure, but it looks like it), so a 1/8" FPT to 1/4" FPT union will be needed as well.
Ideally, I'd be getting an I4TC18FS6QD also available. This is an 18" hose terminated by a 1/4" nipple as well, but with a 3/8" female flare at the other end. Another type of union would be needed to fit this to the Weber 1/8" MPT, but it looks doable, as long as I can get a hold of this hose. I need to find a distributor with stock! The hose on the Weber would provide some measure of "flexibility" and not "stress" the Weber connection with the weight of the Quick-disconnect coupler.