How much generation of hamster's family it takes to get first bad mutations
Everybody carry deleterious mutations anyway

Your two hamsters most definitely already carry a number of deleterious alleles at segregating sites (segregating sites = polymorphic sites). Also, I will not talk about inbreeding depression in your population coming from only two individuals which will be very important but I will only answer your specific question. In short, I will assume your are not interested in segregating sites...

Probability of an offspring to carry a new deleterious mutation

The vast majority of new mutations are deleterious (Shaw et al. 2003). I don't know of estimates of mutation rate in hamsters but in mammals (and in all eukaryote species anyway generally speaking; e.g. see this post and this post or directly bionumbers#mutationrate), the genome-wide mutation rate is of the order of 1 to 100 mutations per offsprings.

If the couple of hamsters have $x$ offsprings, then the probability to have at least one new mutation in at least one offspring is given by the cumulative exponential distribution

$$P = 1-e^{-U x}$$

, where $U$ is the genome-wide mutation rate. With $U=50$ and $x=1$ (one offspring), this probability is $P = 1-e^{-50} = 1 - 1.9 \cdot 10^{-22} ≈ 1$.

In short

In short, to a fair level of approximation, every new offspring carry at least one deleterious mutation that was not present in either parent.