The main objective in modern physics research is to find a way to unify quantum mechanics and general relativity thorugh a series of theoretical approaches called "quantum gravity theories"
But what if quantum mechanics and general relativity are not connected/related? What if we never find evidence of such connection?
The main objective in modern physics research is to find a way to unify quantum mechanics and general relativity...
Not quite. This is certainly an area of modern theoretical physics research, but the majority of theorists work in areas completely unrelated to quantum gravity.
...what if quantum mechanics and general relativity are not connected/related...
The point is that they have to be, at least in some way. Quantum mechanics and general relativity cannot both be correct.
To see this, look at the Einstein field equations (in natural units):
$$G_{\mu\nu}=8\pi T_{\mu\nu}.$$
The left-hand-side tells us about the geometry of spacetime, and the right-hand-side tells us about the matter and energy in the theory.
The fundamental thing to see here is that the right-hand-side is an intrinsically quantum object. The energy density of matter is something that is subject to the experimentally verified laws of quantum mechanics. Thus, since, in general a quantum object cannot equal a classical one, one must either find a way to make the left-hand-side quantum (quantum gravity) or make the right-hand-side classical (i.e. throw away quantum mechanics in the problem).
A really intuitive way to see this is to consider a simple but instructive thought experiment: imagine an uncharged massive quantum particle which is in an equal superposition of two positions: $x_1$ and $x_2$ in a perfect vacuum, which you do not observe the position of (otherwise you would break the superposition). Then, you introduce another similar particle, that you know the position of perfectly. You can then measure the gravitational force between the two particles by seeing how your new particle moves. However, by measuring the gravitational field that your new particle is moving in, you are essentially measuring the position of the first particle. Thus, you have to reconcile the following observation:
By measuring the gravitational field of a particle, you measure the position of the particle.
In modern language, one would say that the state of the gravitational field is entangled with the position of the quantum particle. For such a statement to make sense, the gravitational field must, in some way, behave quantum mechanically.
The idea that gravity is quantized is an almost inescapable consequence of the fact that fundamental particles behave quantum mechanically, and that these particles produce gravitational fields.
