No, this is not how herpes, or any STI, is transmitted.
If someone has oral herpes, to transmit it, they need to either kiss someone, or give them oral sex. It takes direct contact - not indirect contact like fingers - to transmit.
If someone has an actual cold sore, and gets fluid from the sore on their fingers, it might transmit, but for sure, without a sore present, it won't transmit this way.