I've read in old books (100 years, plus) about alcohol "blow lamps". The connecting agent referred to was a form of silver solder. The operator's breath created a small blast of sufficient intensity to create a high temperature. Presumably, this was early brazing.
Hard soldering (silver soldering) is ancient, a trade of the jewellers and goldsmiths.
Theophilus ("On Divers Arts", 12th century) describes brazing. His description is for making rings of iron by wrapping with copper (also with bronze) and then heating in a forge.
Brass (at first it was termed "latten") isn't a common medieval metal. It was made by an expensive cementation process from copper metal and zinc ore, as zinc itself couldn't be reduced to the metal. In the 16th century, brass began to be produced by direct alloying of the two metals. Agricola (De Re Metallica) describes this as a relatively new process and in the UK it began at the well-known Tintern Abbey wire works. As brass is ductile, wire making was an early demand for it.
Throughout the development of firearms, brazing was a common way of joining iron or steel pieces.
"Modern" brazing dates from the popular use of mild steel (later
19thC) as a replacement for wrought iron. Steel is relatively harder to forge weld, so this encouraged alternatives.
The Romans had a significant brass and bronzeworking tradition. I imagine they used it for brazing too, but haven't checked the references.