Brown is being held back as part of "a long-term plan", while Adcock, named as a co-leader on Wednesday, has a slightly delayed start after off-season hip surgery, so they will both miss the matches against Gold Coast and Hawthorn.
Newly appointed vice-captains Daniel Rich and Tom Rockliff will take the reins in the two matches at Metricon Stadium.
Brown said he would return in round three of the NAB Cup against Collingwood at Etihad Stadium on March 9.
"This has been a long term plan, to play the last couple of NAB Cups," Brown said.
"(I'm) feeling really good at training, everything's on track."
Adcock said he would return a week earlier, against Greater Western Sydney on March 2, and planned to play every match after.
Both played in the Lions' intra-club trial last week.
Voss said Saturday night was a great opportunity for the 22-year-old pair of Rich and Rockliff to show their leadership wares.
"The development in that in itself is going to be valuable for our whole group, not only now, but in the future," Voss said.
Voss said the new captaincy structure at the Lions was not necessarily a long-term plan and would be assessed at the end of the year.
"We're hoping that more (players) step up," he said.
"If we are in a position where Browny can step down and continue to play, that's a good thing. He's carried the load for a long period of time, it's time for the rest of us to step up and join him."
