Do the four brothers in TMNT train in other weapons?
Well, Thaddeus Howze's answer already established that, yes, they did. There are other examples from the early comics as well.
Eastman and Laird's first sketches of a ninja turtle (later cleaned up and inked, as seen here) were both with nunchucks... awkwardly tied to the back of the arms for support rather than ready to be used as weapons. As they began to realize what a truly fun idea their truly ridiculous idea actually was, Eastman sketched and Laird inked this image
Put them into Daredevil's origin story, given them a mentor, and give that mentor a reason to restore his honor and that's the entirety of Issue #1 right there. Everything else grew from the rapid success of that first issue and the decision to run an entirely separate kid-friendly universe for the first TV show.
In other words, the turtles have been identified by their weapons longer than they've even had identities. The identities came from the weapons first, before they even had a story, and have continued on through every iteration of the concept. The TV show used color-coded headbands but the original series used all red masks in a black-and-white medium: having the turtles use their different preferred (or possibly only-available) weapons was the way you could tell them apart at all.