It is asking for a favor, so it is politely phrased as a question ("Could your dad ... ?") instead of a request. It means "I want your dad to talk to them, and tell them good things about me." For example, suppose you are trying to get a job as a lifeguard at a swimming pool. You have submitted a job application. Rhoda Jones makes the hiring decisions. You have never met Rhoda Jones. Your friend's dad knows Rhoda Jones and talks to her all the time. You hope that your friend's dad will have a conversation like this: Friend's dad: "I hear Ozan applied for a job with you." Rhoda: "Yes, I have his application. What can you tell me about him?" Friend's dad: "He worked for me last summer. He was reliable. I would trust him to do any job." Rhoda: "Thanks, I'll keep that in mind."

Hi Ozan, It means to make a recommendation - it can either be a recommendation for a person or an idea or plan. So "could you ask your dad to recommend me to his boss?" or "Could you ask your Dad to put a good word in with his boss" is something you might say if you wanted to work for the dad's company.