If the bleeding has stopped then all should be well & you needn't do anything further. I've done this twice to one of my boys who is a squirmer at nail trim time. I always stopped the bleeding with a styptic pencil like men use for shaving cuts. But I felt bad & so now I wrap him in a towel, exposing one foot at a time while he munches on a long stalk of parsley, his fave treat. I get 2 or 3 nails cut per stalk of parsley. He does not seem to notice I am nail trimming while he is munching - utter food focus. His nails are black except for one foot so I have no idea where the quick is. Regularly once a month I just trim a small amount with human type clippers as they do not open too wide. I still keep a styptic pencil on hand but have not had to use it since I started the towel & parsley.
My other boy has to go to the vet every month for dentals so I just let the vet also do the nails at the same time.