If I wanted to convince somebody I was competent, I might start by admitting that there are many more things I am unskilled at or simply don't have personal experience.
This could causes the person to change hats.
Before, they were defense. They were trying to find how I wasn't competent. By me telling them that I'm not competent at everything, they may realize that everybody has strengths and weaknesses and at a minimum, some basic self-awareness is part of my tool collection.
Then, I would define terms.
What does it mean to be competent? Does it mean excellence? Word-class? Or does it mean can show up and follow basic human tasks within a margin of error?
Defining terms helps you get on the same page and more quickly reach agreement. Without a shared understanding, there's always room for miscommunication.
Then, I would define the scope.
Once the terms are agreed upon, I'd ask questions to understand the scope. Scope is not quite the same as terms. It's a refinement. If you asked me if I was a good runner, I might say I can run a six minute mile. If the scope was extended to 100 miles, I'm suddenly not a competent runner anymore. A runner who runs 5k and a runner who runs 160k are not trained the same. The level of competence is different.
Next, projects and process.
Now, I've defined terms and agreed upon scope. A competent person has experience so I would invite questions about my personal projects. Instead of telling stories of my life and work experience, I'd invite them to ask me some specific questions. I'd offer to elaborate to discuss what I learned on a project and add details into how my process works.
Add personality style.
With these items discussed, a big factor is the simple addition of their preference for my style. Do they like how I communicate? Would I be somebody good to work with? Do I appear more or less competent than they are? These objective accomplishments and relative comparisons would help form an image of competence in their mind. If it meets or exceeds expectations, then I've convinced them of my competence.