Given a = bc, with a known integer a, is it possible to find all b and c values that are integers quickly without testing each b and c value?
As an example a = 194920496263521028482429080527, is it possible to quickly find integer values for b and c?
This is a problem of integer factorization. From the factorization of an integer, it is mechanical to write every possible pair of two integers whose product is the given integer.
The method you mention in comments, "a quicker way than individually testing all integer values of $b$ and $c$" is similar to, but less efficient than trial division, trying each integer in $[1,\lfloor \sqrt{n} \rfloor ]$ as a candidate for $b$ and determining whether each choice of $b$ makes $a/b$ an integer or not.
There are much faster methods than trial division. There is not a method that one would consider "fast" for integers of unlimited size.
For your particular example, $$ 194920496263521028482429080527 \\ = 289673451203483 \cdot 672897345109469 $$ is the only product giving that number that does not have $b=1$ or $c = 1$.