Having signed a two-year deal until May 2015, the retention of the 1872 Cup - awarded to the aggregate winners of the derby double header in the RaboDirect PRO12 - which Glasgow have won in each of the last three seasons, is high on Kellock's agenda.
"We have been shaping up over the last two or three seasons to do something special for the game in Scotland and compete with the best teams in Europe," Kellock said. "I want to be at the forefront of the push for success. Winning a trophy against these sides is something I desperately want to happen for the game of rugby in Glasgow."
The second row and captain will end his playing career at Glasgow after signing the new deal, which includes options to keep playing as well as an ambassadorial role with the Warriors and the Scottish Rugby Union.
The 31-year-old and former Scotland skipper joined Glasgow from Edinburgh in 2006 and made such an impression he was immediately named captain, a position he has retained in the seven seasons since.
Kellock's Scotland team-mate Tim Visser, who is in his fourth season at Edinburgh, is eager to claim the trophy for the first time by the end of the second fixture at Murrayfield on December 29.
He said: "It would be good to finally get my hands on the cup for once."
Visser is one score short of equalling the all-time PRO12 try-scoring record of 48 in what will be his 65th appearance and believes success against the Warriors could kick-start Edinburgh's season.
"It's now a vital part of our season, especially now we're out of the Heineken Cup," he added. "Picking up two wins in the Rabo in the last two games has pushed us a little bit up the table.
"Being able to pick up two wins would be incredible. It would put us in a very good position."