You have two options:
Go to the tri on the hybrid. I’ve seen entry level tris and duathlons with people on mountain bikes. Particularly the enticer and shorter distances. If you like it you can think about getting a road bike. (I would not suggest a TT or tri specific bike unless you are very serious about doing long tris. A road bike will make it easier to train on, ride in traffic etc.)
Put drop bars or clip on aerobars on the hybrid. This usually does not end well, as it the frame geometry of hybrid makes will usually make it uncomfortable to ride with drop bars or clip ons. In particular the top tube length and seat tube angle will be too different to make it work. It is also expensive to put drop bars on, the brifters alone will cost a couple of hundred each. It is also a pain to get them to work correctly with V brakes.
My suggestion, depending on how much money you have, would be to put slick tyres on the hybrid, lock out the shocks if it has a suspension fork (or buy/borrow a plain fork) and just go on the hybrid. I would not bother with clipless pedals or anything like that until you are serious that this is what you want to do. Apart from the learning curve associated with them, wearing your running shoes will make the transition much quicker. To get kitted out on a road bike with pedals, shoes etc can be quite expensive. So make sure you want to do this before you spend too much money.
In any case, in my experience the short tris tend to be won in the run anyway. Most triathletes I know are runners who started cross training for whatever reason. Most of the tris I’ve been there were not many people who were strong on the bike or in the water. But they would destroy me on the run.