The TS-7970 includes an HDMI 1.4 port which supports EDID for automatically configuring the video modes on your monitor, and HDMI audio. In Linux this will be /dev/fb0. The HDMI is capable of outputting up to 1080p60.
Under either distribution the mode will default to the largest and highest refresh rate compatible with both the monitor and the i.MX6. You can override this in yocto with xrandr, or under debian/ubuntu/yocto with a kernel cmdline change. Hold SW1 when power is applied and the board will stop at u-boot. Run these commands to override the EDID settings and use 1024x768M@60.
env set cmdline_append "console=ttymxc0,115200 rootwait ro init=/sbin/init video=mxcfb0:dev=hdmi,1024x768M@60,if=RGB24" env save
HDMI to VGA/DVI-A adapters are possible to use with the TS-7970, but the port protection chip strictly follows the HDMI standard. Some adapters are known to violate the standard and try to pull too much power off of the HDMI port. For an adapter to work it should accept a separate power input such as a microUSB port. The TS-7970 can still power the adapter through USB, but the HDMI header should not be used for sourcing current.