Hi Adam, welcome to the forums!
No, unfortunately that is not possible out of the box.
You will need to do some customization for that.
You can either use some javascript or jQuery on the page that will not let the counter go higher than one.
Or you can try a template override where you limit the input field to only allow the number 1, or even eliminate the field or make it read-only. You can find further info on how to create template overrides in our Themer’s Guide.
Or you can try to use the WooCommerce products page to sell the tickets and have a WooCommerce extension that limits the max number of tickets that one can purchase.
At the moment these are the ideas that come to my mind. Hope this helps you get started.
Let me know if I can help you with anything else.
Cheers,
Andras