Martinez met Carrick and senior club officials on Thursday, two days after suggesting United needed greater investment to compete at the highest level following their 2-0 Champions League defeat by Atletico Madrid.
The centre-back later apologised for his remarks and Carrick sought to downplay the episode when he spoke to reporters on Friday.
"We've moved on from that. It was one of those things. It's really not a drama for the group," Carrick said.
"The squad have trained really well and they are in good spirits. There is a freshness about the group...
"He's part of the squad (for Sunday)."
The Argentina international had singled out director of football Jason Wilcox in the post-game interview, saying United needed more signings to compete with their rivals and build a squad capable of challenging in Europe.
Martinez made the comments at a difficult time for United, who sit 13th in the Premier League table after winning only one of their opening six matches in their worst start in 40 years.
The manager said the club had accepted Martinez's apology.
"He's released his statement and realised he made a mistake. That's enough for us to move on with," Carrick said.
"I understand the emotion. We ask for passion. There's a way of doing that and that's been acknowledged. But we need to carry that passion in the right way."
Carrick, appointed permanently in May after overseeing the club's revival last season, has come under pressure following a poor start to the campaign, but said belief within the dressing room remained intact.
"I don't need results. We all need results. It's bigger than me... it's not about any individual player. It's about a football club," Carrick said.
"There's a lot of people saying a lot of things. That doesn't mean you know what's going on here.
"That doesn't distract us from what we are creating here as a group."
United travel to seventh-placed Leeds seeking their first victory in more than a month.
English Premier League
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