The short answer is, yes. You can become a programmer without earning a university degree in computer science or software engineering.
The longer answer is a bit more complicated. Starting your career without a degree will mean doing quite a bit of self-study and self-training. You need to be a hard worker and highly motivated to succeed.
The truth is, this field is big (there are hundreds of programming languages!) and it’s changing all the time. Even with a degree, you’ll spend a lot of time studying new tech and teaching yourself new skills.
So this begs the question: If you don’t want to go to university, what’s the best way to get your programming career off the ground?