Can a filling on the bottom tooth cause pain to the tooth next to it?

Yes. Teeth adjacent to any restoration can feel like they have been treated also. The nerve for both teeth enters the brain together and the brain has difficulty discriminating which tooth has had a new filling.
Maybe. You might be getting referred pain. Or your senses may be just off of one tooth. If the filling was done recently it is possible you have a high spot on the filling, . Too mane scenarios to answer...See your dentist.