So sometimes in the middle of completely normal driving conditions (e.g. stopped at a red light after driving 30 mph on a flat street), with a fully charged battery, HV charge mode will just pop on.
HV charge mode is likely not involved in any of this. When this happens do you see an indicator on the instrument panel that says HV Charge? If not then this has nothing to do with HV Charge and is probably just normal engine startup which can happen in certain situations while driving in EV mode.
One common situation is the engine will start up if you briefly press the accelerator hard enough while driving in EV mode. It's easy to not immediately notice that this has happened because in most cases after the engine starts it just idles for a couple of minutes, and so you may not hear it right away above the normal wind and road noise and you don't realize the the engine is on until a minute or two later when you notice on the display that you are no longer in EV. This creates the impression that it started randomly because you don't realize that the engine actually started a few minutes earlier when you pressed the accelerator. Also you are more likely to hear the engine only after you have come to a stop, then when you hear the engine running it creates the impression that it just came on when it has actually been running for the past couple of miles. Usually after a few minutes of idling the engine will shut off on it's on.
Another situation is when you have a full battery and you do any regen within a few miles from your house, like stopping at a stop sign near your house. Since the battery is full there is no place to store the extra electricity produced by regen so it starts the engine as a way to dissipate the energy. Again once the engine has started it will idle for a few minutes then shut off.
The third and less common situation is something called system check, where the engine comes on and remains on for a much longer time, like ten minutes or so. A few people have said they have experienced that when it goes into system check the engine never shuts off their entire drive. But this is less common than the first two situations.