It is said that speed of an induction motor depends on its frequency and torque depends on current.
Induction motors are mechanically simple, but theoretically complicated beasts. I think the non-attributed "them" that you are citing are oversimplifying.
Let’s think an induction motor runs with constant load and speed. When I increase its current (by varying voltage) but keeping the source frequency constant (by vfd) what happens ?
Then you start operating it outside of the parameters of whatever oversimplification is being made by your non-attributed "them", and "they" are no longer correct.
Like most motors, induction motors generate torque because a magnetic field in the rotor works against a magnetic field in the stator. What makes an induction motor complicated is that it's rotor field is generated by a winding whose current is induced (hence "induction motor") by the magnetic field of the stator. This current varies by the amount of slip (look it up) and the strength of the magnetic field generated by the stator. The relationship between these is complicated enough that it can take more than one book chapter to fully describe -- and it's been a while since I've studied the subject, so I'm going to shamelessly gloss over any details.
I do know that variable frequency drives are designed to more or less hold the motors at constant slip. I say "more or less", because I suspect that there are all sorts of trade-offs between slip and efficiency, and between the expense of the VFD and the difficulty of maintaining an intended slip.
Part of this is that a VFD doesn't just control frequency -- it controls the voltage to the motor. It's constantly juggling both to get that constant slip and maximum efficiency at the desired output frequency.
So here's what happens in your scenario:
Assume that your VFD is (A) capable of driving the motor at maximum efficiency, and (B) will allow you to fiddle with its settings. These are both pie-in-the-sky, because (A) humans built the thing and we aren't that good (although I wouldn't be surprised at a VFD that can come within a percentage point or two of that goal, and (B) the humans who built the thing don't trust the humans that'll use the thing not to screw things up. But I digress.
You set the motor up to run at maximum efficiency at some speed. This is happening because you told the VFD what you wanted, and it's got multiple control loops that are insuring just that. You then tell your VFD "OK, lock on to the frequency you're going, but increase the motor voltage".
When that happens, the stator current and magnetic field increases. This increases the current in the rotor, so the motor speeds up a bit. I say "a bit" because a typical induction motor that's perking along happily probably has a slip of around 3-5%. The slip goes down, but it won't go below 0%, so the increase in speed will be slight. I can't tell you the details, because I'm not a motor expert -- but remember that the VFD was doing a good job of keeping the motor at maximum efficiency.
At best, the motor efficiency will stay close to what it was before. Energy conservation laws demand that the real power to the motor will stay constant. This, in turn, demands that in-phase component of the motor current will go down. So you need to go back and check with your un-attributed "them" to see what simplifications they were making.