I used a lesser genetic mutator on one of my pets and he did not get any new genes or mutations. It says he was mutated but nothing showed up? Does it only show in his children?

Lesser genetic mutators add either a visible or carried trait at random to a completely empty trait slot. Carried traits cannot be seen in a pet's genetics unless it has been genetested (but they can be passed on per usual, rng giving.)
You can find plenty of free pet testing services in the Groups, Guilds, Clubs, and Services forum.

Thanks so much!