Does gravity become infinity where the center of gravity of a mass lies?

Gravity pulls towards the center of mass. Simple Example;

Acceleration due to the gravity of earth on its surface; g = GM/R^^{2}

Acceleration due to gravity of earth at its center; g = GM/R^^{2} = INFINITY, where R=0

This equation doesn't work to calculate acceleration due to gravity, because you're using it in a case where nothing is being accelerated. There is no 'm' in your calculations, therefore there is no acceleration. You can see this because:

g = GM/R^

^{2}=nothing

There is no R, since there's no 'distance between' anything, or 'radius between' anything if you like, since there's only one object. Also, even if R=0, anything divided by 0 is nothing, not infinity.

So no, gravity is not infinite at the centre of mass, nor is gravity necessarily infinite at the centre of black holes. The Earth won't get sucked into itself, that doesn't make sense. For the Earth's mass to become a black hole, the Earth would have to be crushed to the size of a peanut. Then the density is so great it collapses in on itself to become a black hole.

tl;dr - your equation is wrong