Engine starts and die after couple of seconds

You let the magic smoke out! ​

 

But seriously, you need to take it to a dealer, or someone with Toyota diagnostic software to have the trouble codes scanned.
It might be as simple as a burned out inverter fluid pump , and related fuse

 

Sorry; a bad fuel pump isn't going to cause smoke in your engine compartment. The source of the smoke needs to be tracked down. Your mechanic is making a good guess, because what you've described is the EV motors in the transmission cranking the engine. Your engine isn't running or starting.
Be careful, an old, high mileage car like that can easily deplete the traction battery - so you may also need a traction battery in the near future too.

Good Luck....

 

There is a cooling water pump near the left headlight in a gen 2 that is not a rare point of failure. Often it lets out a very stinky smoke when it fails.

Sometimes only the pump fails, and the car can still be driven, but the inverter may overheat.

Other times, the pump fails and blows the AM2 fuse in the fusebox. Because there are other parts of the car also powered from that AM2 fuse, when it blows, the car will not work.

Just replacing the AM2 fuse is no good if the pump has not yet been replaced; the new fuse will just blow again.

There is a handy connector that's unpluggable to prevent a bad inverter pump from continuing to blow the AM2 fuse while you arrange for replacing the pump.



With that connector unplugged, there is no coolant flow in the inverter electronics, but gentle driving for short distances will probably be ok until the pump can be replaced and the connector plugged back in.

 

The car was doing what at the time? Presumably running but if you were sitting still the motor should have been off.

Smoke was grey, black, white? How long did it last? Where exactly did it come out? You say near the fuse box but presumably the car has a hood, and even if it didn't, the hood is not visible from the driver's seat, at least for me. Also, this is a right side driver vehicle? Most of us have left side driver cars, so things under your hood may be in different places than ours.

No codes?

A burned out inverter pump might smoke but it will generally not turn off the motor instantly, it will light up the dash and throw at least one code. It might turn off the motor if the ambient temperature was very high or the car was under a lot of load.

If there is a burnt smell under the hood try sniffing around to find whatever it is that burned. It wasn't the fuel pump, that is inside the gas tank in back.