Can I increase the generator output power by increasing its speed?

IF you are connected to STIFF grid, yes. If your generator is connected to a soft grid, it's different.

By definition, if this generator can impact the Hz or V of the grid, it is a soft grid to this generator's control system. For example, two 10MW generators and no grid connection. One generator can "push" the other around up to some point. It is NOT size dependent: you can parallel a 10MW with a 10kW and to the 10kw, it is a stiff grid. To the 10MW, it is a soft grid.

On a soft grid, fuel still is proportional to torque. It is just that torque produces an increase in speed / Hz. This happens a lot in distributed power generation.

Just like on a stiff grid, on a soft grid excitation produces voltage. That voltage might increase VARs or it might increase V, depending upon PF and other generators on the grid.

RPM = Hz so you can't change the speed without changing the frequency in the AC world. A 20MW generator, for example, is going to require a prime mover capable of approximately 32,000 HP, (Horse Power). The prime mover can be anything you want that can produce the required rotating power, reciprocating engine, combustion turbine, steam turbine, water turbine, etc.

The power output of the generator can be reduced by restricting the power output of the prime mover. This will cause a frequency drop when the prime mover becomes overloaded.

The AC power output of the generator can be restricted by reducing the current set-points on the overcurrent relays. This will cause the generator output breaker to trip when the load limit is exceeded.

The AC power output of the generator can also be restricted by limiting the excitation voltage to the generator. This will cause the generator output voltage to fall when the load limit is exceeded.