Former Hoops and Scotland midfielder Collins was confirmed in the post on Tuesday, with youth coach John Kennedy also set to move up to a coaching role with the first team.
While at Hibernian in 2007, Brown was in a dressing room which agitated against Collins when he was Easter Road boss, with a delegation of players widely reported to have visited then chief executive Rod Petrie to voice concerns about training methods and team selection, despite the former Monaco midfielder having brought the Scottish League Cup to Leith.
Deila was speaking before he kicked off this year's Glasgow Taxis Outing for Sick Children, which was supported by Celtic with a £7,000 donation, and he was confident that time has been a healer, although he is passionate about opinions being expressed.
"I talked with Scott about what I'm thinking of doing and we have agreed on everything, "said the Norwegian.
"If they had something in the past, they are grown-up people and I know there is going to be no trouble.
"It is the same with John and me. We need strong personalities, we need many opinions and we have to handle them in a good way and then we will progress and produce good results for Celtic.
"Scott is a fantastic captain so of course he will be captain this season."
The former Stromsgodset boss, appointed Celtic manager this month following the departure of Neil Lennon, was keen to stress he had been behind, rather than simply rubber-stamped, the recruitment of Collins.
"It was, of course, my choice," he said. "We had a lot of options. I am very pleased to get him here.
"It was important for me to get a guy who knows Scottish football and who is a good name for Celtic.
"When we talked we didn't stop for a long, long while so I think we will have many good discussions.
"He was a very good footballer, he has a good knowledge of the game, he knows Scottish football very well and has been a professional in France so he knows the European culture and the standard of the Champions League.
"He has similar views to me and he is a very strong person and I want strong people around me who can fight with me so we can improve as a team."
Deila is relaxed about the fact his number two has a bigger profile.
"I love strong personalities," he said. "I love knowledge around me.
"If you are a good leader you can handle that. I want staff who have a good knowledge of the game, and opinions."
The Celtic squad return to pre-season training next week ahead of their first Champions League qualifiers next month, and they learned on Wednesday that they will begin the defence of their Scottish Premiership title with a trip to William Hill Scottish Cup winners St Johnstone.
The Hoops were drawn at home to Partick Thistle on Saturday August 9 but, owing to the Commonwealth Games, the clash has already been postponed.
As such, Deila will first enter the dugout for a league match on Wednesday, August 13 at McDiarmid Park.
He has begun his search for new players, with former Hearts and Sunderland goalkeeper Craig Gordon in the frame, but is unlikely to return to Norway to plunder his former club.
"I don't think so," Deila said. "If you are going to play for Celtic you need (to be) Champions League standard.
"Norway is far away from that. But players can come to Celtic from all over the world.
"I have good people around me and things will happen during the next month.
"We are going to go after talent who have the quality or can get it.
"We are talking about Gordon and we will take it from there.
"He has been a fantastic goalkeeper, he has been injured for some time but he looks good now."
Looking to his first Premiership game, Deila said: "It is going to be a good game, St Johnstone are the cup winners.
"But first we have the Champions League (qualifying stage). We will do everything we can to be well prepared."