I will join longneck, you need to give to give this one to a pro.
If you do not understand about back-feeding a circuit, already, it is too easy to kill someone. I say this without exaggeration.
The biggest problem with running a generator with a transfer switch is that you have to wire it into your main panel.
For basic protection of life, (as well as the law,) it must NOT be possible to have your generator and the power to your house on the same wire, at the same time.
This is not a case of, just for a few seconds, or I would not do that, or I will triple check a written list.
It must not be possible under mechanically, electrically, on with quantum physics.
The reason is that if you put power on the 'dead' wires to your house, it is possible to kill someone a mile away, or more.
Emergency workers expect down lines to be dead. Yes, they do check first, but that does not mean that the situation can not change while they are working on them.
Transformers work backwards, low voltage at your generator (240) quickly becomes high voltage (thousands).
And sometimes people, and pets, or cars, just do not even see a fallen power line.
This is a case of some money to a pro vs life.
You should replace the garage outlet with an inlet, such as this (Amazon, as an example).
This way, you can use a standard male-female cord, either a standard extension cord or one made with the correct gauge of SO cord [Thanks, @EdBeal]. You avoid using a male-male cord which is so dangerous.
Best Answer
What you describe won't work, and it's not safe for you (or anyone else) to do what you describe.
If you have a single phase 120 volt generator and you need to supply power to a 240 volt split phase load, the safest way is to use a transformer with a 120 volt primary connected to the generator and a 120-0-120 volt secondary connected to the appliance.