Joe,
I've never tried it myself, but I sure don't remember many stories of success running a buzz box off of a small generator, and I'd classify yours as a very small generator... they just pull too many amps. Arc strikes are typically very difficult at best, and if you do get one lit up, you can count on popping the generator's CB's in short order.
I'd say you would probably need at least a 6000 watt generator to run a buzz box with any chance of success on any but the absolute lowest amp settings. 10,000 watts would be a lot better, especially for what you're wanting to do. With your generator, I think you might be able to run small 1/16 and maybe 3/32 rods up to the 60-80 amp range, maybe just a tad higher if you're lucky, but I really don't hold out much hope.
When you're welding your 1/4 inch, you'll typically be using 1/8" rods in the neighborhood of 135-140 amps. Arc voltage will probably be around 22 volts, which calculates to at least 2970 watts. I'm just guessing, but if the transformer is 85% efficient, that bumps it up to 3494 watts on the input side, which at 240 volts, is 14.56 amps required from the generator. So, with just "steady state" welding, you're right at the ragged edge of your generator's capability...
You can surely try it, but I'd say your chances for success are very slim at best.