This is a poisson process so we have the following properties:

Start from the usual formula for conditional probabilities

The denominator is easy using property [B].

The numerator can be re-written as follows:

using property [C], the events and are independent, and hence

Both of those terms follow a poisson distribution. (unless k-m is negative, then the probability is zero).

So the final answer is:

Where:

you can subsitute the formulae for the PMFs to finish. Dont forget that the probability is zero if m > k