Is there an integrated terminal option in file managers such as Nautilus in Ubuntu?

This is the standard behavior of dolphin, the stock KDE file manager (which is by itself a sufficient reason to use KDE). However, to use dolphin under Gnome, I think you will also need to install Konsole and it feels a little out of place on Gnome.

An alternative seems to be Nautilusterminal 3. Though, I have not tested this myself, as I am a KDE user. You can install it by:

sudo add-apt-repository ppa:flozz/nautilus-terminal
sudo apt update
sudo apt install nautilus-terminal

Since, as of writing this, no Ubuntu 19.10 version is included in the ppa, you can use pip:

sudo apt install python-pip python-nautilus python-psutil
python2 -m pip install --user nautilus_terminal

The project is hosted here. A deb package can be found here.