After extensive searching the InterWebs and finding a lack of documentation on how to install TShark in a silent/unattended mode, I came across a related StackExchange post by Franck Dernoncourt that got me on the right track. Here is what I came up with for Debian:
DEBIAN_FRONTEND=noninteractive apt-get -y install tshark
For a pet project I am working on, I am creating an install script to install several security tools onto a virtual machine and got stuck when TShark popped the following Qt screen:
My goal is to have everything installed and properly configured without a user's involvement. There are several resources that show how to run Wireshark (TSHARK) as non-root, but that is not what I wanted. I was ok with the default "<No>" but wanted to avoid being prompted for it.