How to use custom page templates in a theme that hides page template option?

It could have been disabled in the “screen options” dropdown at the top of the page.

But, if it hasn’t, and you need to only affect one page, then you can use the naming convention of “page-(ID).php” or “page-(slug).php” outlined here, however this can cause frustration if you try to change a template after the fact, so please use with caution.