Induction motors run on a design Volts per Hertz ratio. You didn't specify the rated voltage of the motor, but if it is 50Hz, I'll make the assumption that it is a 380VAC three phase motor. 380 VAC at 50Hz is a ratio of 7.6. Almost all 380VAC rated motors have an insulation system that can safely handle 460 VAC, which is another motor standard voltage. 460 at 60 Hz is a ratio of 7.666, so yes, the speed can be safely increased.
Now, you need to consider some other factors about induction motors. Almost all induction motors under 75KW or 100HP can be safely run up to twice rated speed, but in order to do that, you lose torque. When running over base speed, the stator voltage is kept at the maximum (380 or 460 for example), but the frequency increases. If you were to compare the operation of an induction motor from 0 RPM up to base speed, and then to the extended speed, it would be almost identical to running a DC motor from 0 to rated speed with full field, and then decreasing the field current to increase the motor speed. From 0 to base speed, both operate in the constant torque region. Past base speed, they operate in the constant power (KW or HP) region.
To answer what I think is your actual question, no, do not run a 50Hz motor at an increased voltage unless you increase the frequency to 60Hz to keep the same V/Hz ratio. Increasing the voltage without increasing the frequency leads too excessive winding heating and also reduced cooling capacity from the internal fan (or just the rotor turning).
Other considerations can come into play depending on the mode your VFD is running in, but the above are the basics.
There is a good possibility that the VFD that you have can be adjusted to provide an output of 150 Hz at 240 V with 480 V input. You may be able to get it to operate at 1 Hz, but most motors won't operate very well at 1 hz. A good sensorless vector drive may be able to do that. You can probably not accomplish much trying to use an induction motor as a generator.
When operating induction motors above their rated speed the main limitations will be bearings and rotor balance. At some speed, centrifugal force on the rotor may become an issue, but that speed is likely to be significantly higher than the rating of the bearings and the speed at which rotor balance becomes a problem.
With standard motors, you can probably run a 6-pole motor at twice its rated speed, a 4-pole motor at 1.5X to 2X, and a 2-pole motor at 1.25X to 1.5X. If you don't increase the voltage proportionally for operation above nameplate frequency, the motor will not be able to deliver rated torque. You can probably get constant horsepower operation up to 1.5X rated speed. Above that, the torque must be limited to something below the torque that will provide constant horsepower.
Re Question Edit
There is no way that you are going to get an induction generator to give you the frequency range you are looking for.
The best option would to directly use the output of a VFD if you can get an acceptable waveform. An up-to-date PWM VFD will give you a a pretty good waveform. You can probably find one that will go down to 1 Hz or close to that. You could also consider using a VFD with an output filter. You may have difficulty getting good information about waveform quality. It may also be difficult to put together a filter.
For a really good waveform, the best option will probably be a permanent-magnet synchronous generator (PMSG). That will give you a good sine wave with no filtering. You can use a permanent-magnet synchronous motor as a PMSG. You use a PMSG that has more poles than the driving motor, your speed can be lower. You would need to consider the power factor of the load. I don't know how a PMSG will react to a power factor below 1.0. A VFD will have no problem with a lagging power factor down to 0.7 or so.
A wound-rotor synchronous generator (WRSG) can do as well as a PMSG, but you may have difficulty getting an excitation control unit that will work over your frequency range. Most WRSGs in the size range that you need are sold either with an engine or as "head" units designed to be bolted to an engine.
Do you need to put this together yourself or do you have a budget that would allow you to work with a system integrator?
Additional Details
The performance of a PMSG will be similar to the performance of a WRSG that has a fixed excitation current. It will not be possible to adjust the output voltage. The output voltage will vary with power factor and load current variations. With a VFD or any type of generator, the both the output voltage and the speed will be directly proportional to speed.
A VFD will have considerable flexibility to program and adjust the output voltage, both independently and as a function of frequency. Generally, the adjustment would need to be done with the VFD output shut off, but it may be possible to configure a VFD for voltage adjustment while running.
It seems that a VFD will probably be the best alternative. If you don't want to design and build an output filter, there are suppliers of VFD input and output reactors etc. that will very likely be willing to supply one.
Here is a link to some basic VFD information.
Here and here are links to VFD output filter information.
Best Answer
When you're setting up a Variable Frequency Drive (VFD) you will configure things like the min/max frequency, rated voltage, rated full-load amperage, and accel/decel time. Depending on how fancy of a VFD you've got, there could be dozens of other settings (comms., safety, digital inputs/outputs, control methods, etc).
As mentioned in some comments, many industrial motors are manufactured to 50/60 Hz depending on your geography, but this is not true of ALL motors. I believe ships and planes use higher frequency. Wikipedia says that spacecraft, submarines, machinery, and military equipment may be run on 400 Hz. Motors run at 400 Hz can provide similar power for a smaller size.
To answer your question -- 590 Hz out of the VFD is possible, but it is not safe for the operation of a 50/60 Hz motor. In this scenario, you would configure the VFD to 50/60 Hz. If you were using a 400 Hz motor, you would configure the VFD to 400 Hz.