Using this Workflow file you can deploy your Doxygen docs to GitHub pages (in this example)

makeDeploy_Doxygen.yml

name: Doxygen

on:
  repository_dispatch:
  push:
    branches:
      - master
      - gh-pages

# In that case do the job 'make_and_deploy_doxygen'
jobs:
  make_and_deploy_doxygen:
    runs-on: ubuntu-latest
    # which needs the following steps to be executed:
    steps:
      # 1. Checkout current branch of GitHub repository.
      - name: Checkout current branch
        uses: actions/checkout@v2
      # 2. Install doxygen and its auxiliary components.
      - name: Install doxygen and latex components
        run: sudo apt-get update; sudo apt-get install -y doxygen graphviz texlive-full
      # 3. Create the doxygen pages.
      - name: Create the doxygen
        run: |
          git clone https://github.com/metacall/core.git
          mkdir core/build && cd core/build
          cmake -DOPTION_BUILD_DOCS=ON ..
          make api-docs
      - name: Moving Files
        run: |
          mv ./core/build/docs/api-docs ./api

      # Deploy to GitHub Pages
      - name: Deploy
        uses: peaceiris/actions-gh-pages@v3
        with:
          github_token: ${{ secrets.GITHUB_TOKEN }}
          publish_dir: ./