It's not clear what does it mean "didn't trigger", because you don't say how you set up the trigger. Anyway, at first you could just set the scope trigger to "Auto", set like 10mV per division, and look on a running trace if you see anything going on when you activate the welder. If you see something, but very low amplitude, you can take a piece of thin wire and wind a sensing coil with 10 or 20 turns, not just 1. That will increase the amplitude of signal by a lot.
Regarding the direct connection, yes, it certainly can be done, but I don't want to be responsible by suggesting how to do it. Because if by any chance your welder output is not isolated or some momentary arcing produces high voltage, it might not be safe for a scope. I would myself either use a loop / coil approach described above, use a differential probe or check things really well and include at least rudimentary protection on the scope input before attempting. Personally I prefer to be on a safe side whenever possible.