No.
A number of common starship facilities that would be on the Enteprise are not actually seen in the Abramsverse films.
In neither Star Trek (2009) nor Star Trek Into Darkness (2013) do we see a mess hall, a ship's galley, an officer's lounge, or any crew quarters.
Between all the yelling and running around in the reboot films, there doesn't seem to be any time for eating or sleeping.
Apparently, the DVD extras for Star Trek (2009) have undeveloped concept art for quarters. They may also have concept art for other facilities such as the mess, but I cannot attest to this.
See also Richard's answer to Crew quarters on the Enterprise in Star Trek (2009) and Into DarknessCrew quarters on the Enterprise in Star Trek (2009) and Into Darkness.