This idea seems to crop up now and then. I will have to confess the thought occurred to me as a kid in grade school. The AC Clarke “2010” featured Jupiter being imploded by black monoliths and turning into a star.
Even if you get a fusion bomb into the core of Jupiter, which would be tough to do, the explosion would have no measurable effects on the surface. There simply is insufficient pressure to fuse nuclei together. You would need some sort of pressure producing system that is also stable. You might consider putting a small black hole (an Earth mass BH or so) in the core of Jupiter, which might create a zone of imploding material that has nuclear fusion. The rub with this idea is you also ultimately turn Jupiter into a 3-4 meter radius black hole. A collection of Hawking radiating black holes might be better, but their masses would have to be tuned so they don’t absorb or lose mass. This would mean turning Jupiter into a Hawking radiation source as a "star." That is not a stable system. So to play it safe you tune the Hawking BHs to radiate away and then replenish them later. Again, none of this is a small feat.
Of course the idea is to create a sort of mini-solar system with the various moons. So even if you could by some means turn Jupiter on as a star the results after a few thousand years would be to turn the Jovian and Saturnian moons into liquid bodies. These moons would not turn into Earth-like planets.