You should say:
• Who you had the disagreement with
Today, I'm going to talk about a heated argument that I had with my mother a couple of years ago. At the time, I was still in grade 12 and had to make up my mind about what I should major in at university.
• What the disagreement was?
So, I wanted to become a policewoman. However, my mother didn't go along with my plan because she thought it was a dangerous job. She expected me to follow in her footsteps and become a teacher instead. She told me that she had been working in that field for many years and had lots of friends so she could help me get a job immediately after I graduated from university. Of course, I didn't agree with her, so I tried my best to change her mind. My mother was pretty mad at me for not listening to her, then she decided to give me the cold shoulder after that.