I have a 3-year old Trek hybrid. When starting up a hill, I downshifting from 3 to 2 on the front chain ring and the chain falls off. Is it because I'm not pedaling fast enough or an indication something mechanically is wrong?

Does the chain fall off when shifting from the big chainring to the middle one when you are peddling on flat ground?

There is probably something out of adjustment.

I suggest you be more specific. When you say the chain falls off, do you mean it goes to the inside of the small chain ring, or something else?

Which cog are you using in the rear when you make this shift? If you are on the largest rear cog with the chain on the large chainring you should shift the rear a couple of cogs smaller before you shift the front

No

If the chain only falls off when going uphill, it might be something to do with your shifting technique. Maybe try easing off the power as you shift, or better yet, get in the habit of shifting to the smaller chainring before you get started up the hill.
my wife had this problem on her triple. We went so far as to replace the front derailleur with a better one, but she just couldn't get the hang of it. I "fixed" the problem by removing the big chainring and putting a chainguard on, forcing her to ride in the middle chainring all the time.

It's something mechanical, unless you're back-pedaling a little when you shift.

My wife had a knack for throwing her chain when trying to shift going uphill. She insisted there was something wrong with the bike. I rode her bike uphill and couldn't replicate the problem. Took it to the shop and had the bike mechanic take a look at the shifter and derailleur. Eventually they did replace the front derailleur mostly because the shifting was clunky, and they replaced it with a Sora, which shifted better. But even then, it was like clockwork. At least once a ride, she would throw her chain, and we could have to stop so I could put the chain back on the chainring. I think she is more of a masher in her peddling technique. I asked her how she rode up hills, and she said she hung on the biggest gear possible until should couldn't pedal anymore, then she would shift to the smaller chainring.. I told her that isn't how you ride up hills, but trying to coach her up didn't work. Honestly, she probably would be better off going with a 1 x setup, but that was a bit expensive to do on a (then) almost new bike, so we effectively turned it into a 2 x,by removing the big chainring, and also switched up the cassette from 12 - 26 to 12 - 32 so basically she almost never has to use the left shifter. It is a hack, but at least I am not getting my hands dirty every ride fixing her chain.