Well, yes and no...
They DO have a permanent magnet type alternator, and CAN make power without the battery needing to supply anything.
BUT, none of them (the twins) are set up to power the spark that way. On them, they ALL need battery to feed the coil to spark the plug.
On your 125, it depends on the year. The early 70s versions (and the CL's and SL's) needed the battery, same as the rest (well, technically, they could be adapted, but it's not a trivial matter, so I ignore it here). But, in '74 when the XL125 superceded the SL125, they switched over to a magneto-style ignition. On those, they didn't require a battery to spark the plug. The CB125 got the same treatment, I guess...
Looks like a real sweetheart.
Kirk