Layman question here, is it possible for an object to be negatively attracted to a gravitational pull of another object? Like how same pole magnets repels each other, can it happen on gravity too?
No - one of the key differences between electromagnetism and gravity is that the latter only has a single "charge" and everything always attracts.
This holds true even in general relativity, but as long as you are only talking about the mass/energy density of material. In GR, pressure itself can cause gravitational effects, and these can indeed look repulsive, as is the case for dark energy driving the universe's expansion. Even dark energy, though, has positive energy density. If we ever discovered something with negative energy density, well, that would break a lot of things in physics for starters.
Have a look at Why do same/opposite electric charges repel/attract each other, respectively? for an explanation of the attraction between charges (I'm using the word charge in the general sense). It looks long and intimidating at first glance but persist because it isn't really.
For there to be a gravitational repulsion there would have to be negative mass e.g. some object with a mass of -1 kg. Note that this is not the same as antimatter - antimatter has a positive mass just like normal matter. We have never observed any matter with a negative mass, and as Chris notes in his answer negative mass would violate all sorts of conservation laws so it probably doesn't exist.
Negative attraction of gravity may not exist in one sense. It seems gravity always attracts. However, there are many scenarios where gravity forces can be regarded has having both positive and negative values.
I will try to explain.
Gravitational repulsion may not exist but negative gravity is relative. Here is a simple example using Earth. If you take a smaller mass below the surface of the Earth, it starts to have Mass behind it. This is negative gravity in relation to the small object. So the gravitational attraction to the center is negated by the gravitational force of mass behind it. Take the smaller object to say 1 meter from the center of gravity of the Earth and you can no longer use the standard formula where F = (G * M * m)/R^2 . Gravity forces tend to cancel out towards the center. At the very center of Mass of the Earth there should be no net gravity force from the Earth.
I would welcome any links to more data on this.