The former Celtic and Scotland skipper was a teenager making waves with Hibernian when Smith put him on during a 1-1 friendly draw against the United States in November 2005.
The pair would go on to compete against each other when Smith returned to Rangers and Brown embarked on a highly successful 13-year career with Celtic.
But the respect and contact remained firmly in place and Brown will be forever grateful for getting his first international cap.
Speaking following the news of Smith's death, the Aberdeen midfielder said: "It's so sad. Walter was a fantastic man and he was the one who gave me my first Scotland cap many years ago.
"He gave me a bit of advice. He went 'Wee man, go and cause carnage. You are going on up front, go and cause carnage and get yourself a goal'. To be fair, I scored a goal but I was offside. It was a fantastic occasion.
"It was probably one of the first times that I met Walter. He was this huge presence. You knew when he came in the room, everyone went dead quiet, everyone had this huge respect for him.
"I worked with him very closely with Scotland and I've got huge respect for him, especially what he did with Tommy Burns as well, carrying his coffin.
"He tried to bridge the gap between the two clubs, which was fantastic at the time.
"Every time I met Walter he had so much time for every single person."
Smith tried to take Brown from Hibs to Rangers but the midfielder had his heart set on moving to Celtic Park.
"I had a good relationship with Walter, as everyone knows," the 36-year-old said.
"Him and Coisty (Ally McCoist) were there with Tommy with Scotland. It was great banter with the three of them and it was so chilled and so easy to go into.
"It was obviously a hard situation for myself, 17 or 18 at the time and coming into the squad, and everyone made you feel welcome, which was fantastic, and especially the coaching staff.
"I had the chances to go to Rangers. It was nothing on behalf on Walter's part, it was (about) the club I wanted to go to at the time.
"But we still spoke whenever we met. I met him down at Cheltenham a few times and he gave me a couple of horse tips. We still kept in contact that way."
Brown will lead out Aberdeen on an emotional night at Ibrox on Wednesday when the Dons visit on cinch Premiership duty.
"I'm sure both sets of supporters will show their respects," he said. "I think there's a minute's applause and I think that's the least we can do for the man."