How do I set the TERM environment variable for the Gnome Terminal Emulator app?

There are two issues here.

One is how to detect if you're running inside gnome-terminal (or any other app using the same vte widget for the current terminal emulation). Assuming you're on a recent enough Ubuntu, the presence of the $VTE_VERSION variable tells you this. You could edit your .bashrc to contain something like

if [ -n "$VTE_VERSION" ]; then
    # Gnome-terminal and other vte3-based emulators
    color_prompt=yes
fi

The other issue is that for some reason Ubuntu's default .bashrc tries to look for xterm-color which is not really used nowadays. Either the standard xterm (which supports colors anyways), or the more modern xterm-256color are the popular choices, the default slowly moving towards the latter. (gnome-terminal changed its default from xterm to xterm-256color in 3.16, which is present in Ubuntu 15.10 beyond)

If you would like to move forward and already enjoy plenty of colors in several applications, I recommend that you already switch to TERM=xterm-256color, which you could also do in the if-branch above.

Ubuntu's .bashrc is obsolete for using xterm-color. I much rather recommend modifying these bits in the script around whatever $TERM you already have (xterm) or would prefer to have (maybe xterm-256color), rather than the other way around, modifying $TERM to match .bashrc's requirement.

For gnome-terminal

Edit the profile settings in gnome-termninal:

Replace /bin/bash with your shell command and use your parameter for TERM.

You can also edit the entries with dconf. Open a terminal and start dconf-editor:

dconf-editor

Open the entry org/gnome/terminal/legacy/profiles: and select your profile. Change the value for custom-command.