The sip show peers it is a Natted qualification ping (read sip qualify), the qualify=yes, this option is set on the sip.conf file to monitor that the remote sip peer is alive.
sip show peers provides real time information only. It’s primary purpose is not to show the results of quailfy.
Assuming a standard installation, /var/log/asterisk contains log files that show historical information, although most of this is for human use not for machine processing. You may have to set appropriately high verbose and debug levels, and you may have to adjust logger.conf.