Every integer has probability 1 to be divisible by 1. Similarly, every integer has probability $1/2$ to be divisible by $2$. The induction is to consider all prime factors. For the $k$-th prime $p_k$, every integer has probability $1/p_k$ to be divisible by it. Therefore, two independently chosen random integers simultaneously being divisible by $p_k$ is $1/p_k^2$. This means that the probability that two integers are not simultaneously divisible by a prime factor $p_k$ is $1-1/p_k^2$, which means they are co-prime.
Therefore, the total probability would be the probability of the intersection of an infinite number of events. These events are mutually exclusive because the factors are prime numbers.
$$ \prod_{k: p_k \text{ is prime}} \left( 1 - \frac{1}{p_k^2} \right). $$
I am not sure what this number converges to.