The Second Temple was destroyed around 70 AD (or CE for the PC). The Al Aqsa Mosque was apparently built (completed) around 691 AD, by the Caliph Abd al-Malik. Therefore the Templars would have been digging under the Al Aqsa Mosque during their residency in the "Solomon's Stables" area between 1118 and 1127.
Here's a good site with history of Solomon's Stables that places them both physically and historically.
Solomon's Stables and the Southern Gates
The site itself, TempleMount.ORG is an overall history of the Temple Mount and appears pretty good, archaeologically.