It will, but that ground coming from the generator is to ground your equipment to trip the generator breaker if necessary. But you don't want the nuetral of the generator touching that ground after the internal wiring. (It's the same way in your house panel, they are bonded at the panel, but after that you don't want to touch those together. They're doing 2 different tasks.) so ideally you want to run a grounding wire from your panel to the generator ground.
But if your house is old enough with that three prong dryer plug, they may already have the nuetral/ground tied together there.
Next time update the dryer plug to four wire circuit and use a 4 prong plug. Or actually get a transfer switch.