I recently read an article where it mentioned that the solar system could be born out of a supernova explosion (death of a previous star). In case there was a supernova, wouldn't there be a possibility of a neutron star or black hole somewhere close to solar system?
I don't think the article you read meant that the Solar System had been born directly from a particular supernova. Rather, it means that the material that makes up the Solar System (especially the heavy elements, beyond iron in the periodic table) was produced in earlier supernovae somewhere in the universe.
You need the immense pressures found inside supernovae to fuse the heavy elements so gold, silver, uranium. platinum etc. were all formed in supernovae.
As for timescales, the Solar System is about 5 billion years old. The universe is 13.7 billion years old. So it was already about 8 billion years old when our sun began to form. There had been plenty of time for many cycles of supernova formation to occur by then. In terms of distance, our sun orbits the galaxy centre in 200 million years, so we could've made 25 circuits of the galaxy since the sun formed. So any contributing supernovae don't need to be nearby.
Of course, it's entirely possible that there is a nearby neutron star, but that would just be chance and there would be no implication that it had been particularly involved in the formation of the Solar System.