The console on the Drabonboard 410c is typically found on the LS connector. There are two serial ports here, LS-UART0 and LS-UART1, and for most images the console is found on LS-UART1. These UARTs use 1.8v TTL signalling so they typically require an adapter board of some kind. For convenience I'd recommend the 96Boards UART board (https://www.96boards.org/products/mezzanine/uarts/ ) which is cheap and easy to use.
Having said all of that I'm afraid I have not tried the Brillo image on Dragonboard 410c so I cannot 100% confirm that the console will be present on LS-UART1. However it is what I would expect because I have seen other Brillo devices that do have console (and also that no HDMI socket).