Davies was announced as Celtic’s new assistant manager last week, after he agreed terms with the Glasgow club.
Davies, who is still only 31, and Rodgers have previously worked together at both Reading and Liverpool before.
And Brown said: “I haven’t met them [the new backroom coaching team] but I am the same age as Chris so at least we have something in common so far!
"But I have heard a lot about him and when I spoke to the gaffer, he was buzzing about him so I’m sure we will get on well because if the gaffer speaks so highly about someone like that, then you don’t take that lightly.
"They have done really well previously in the game and hopefully they can do the same with us.”
Rodgers has also brought in Glen Driscoll to work as Celtic’s head of performance, as the Northern Irish boss restructures the Bhoys’ backroom team.
Rodgers’ arrival at Parkhead has created great excitement amongst Celtic fans, who are optimistic that he will be an upgrade on his predecessor Ronny Deila.
Rodgers first task will be to guide Celtic into the Champions League, while he will also be expected to help them win their sixth Scottish Premiership title in a row.
Rodgers has already started making changes to Celtic’s team, with both Anthony Stokes and Colin Kazim-Richards departing recently.