Yes, you can serve .pug files with apache, but in this case, you need Node in addition to or instead of your php.
For this reason, I referred to the Pug CLI. With this tool, you can convert the .pug files to .html files.
For example the following project directory:
project-name
| - dist
| - src
| | - index.pug
| | - _assets
| | | - navigation.pug
With the command pug src -o dist you generate:
project-name
| - dist
| | - index.html
| - src
| | - index.pug
| | - _assets
| | | - navigation.pug
But it is like @maruru said. Pug is another language and it is hard to implement it in this state of the project. I used Pug as an example because I'm familiar with it.
There are other template languages with a more html like syntax. A quick google search and I found Twig for php and ejs (with CLI).