Yes, you can check WHM > Daily Process Logs for MySQL activity by set users.
You can run
mysqladmin proc to check the currently running processes (or
watch -n 5 "mysqladmin proc" to have a time delay refresh of it).
You could enable the general and slow query logs as well. We don't have the information on what MySQL version you are using to provide more input on that. You can find out your MySQL version by running this command: