Moyes, now manager at Manchester United, has suggested the careers of the pair could be damaged if they are not allowed to move to Old Trafford.
Everton this week rejected a joint £28million bid for left-back Baines and midfielder Fellaini as "derisory and insulting".
Martinez, speaking after his side's goalless draw with West Brom in the Barclays Premier League, said: "I think my comment is quite clear.
"If managers start talking about players who are registered at other clubs, that is the end of the whole thing.
"I think everyone should look after their own things internally.
"I am sure they have got many issues in the same manner and I could easily come out and give my opinions about these players but I will be more respectful than that.
"Everyone has got a price (but) when you invest money in a player and five years later you get offered the same money after performing really well, that becomes a little bit of a nonsense.
"The last thing I want is to get into a media war with a manager I really respect."
Baines joined the Toffees from Wigan for £6million in 2007 with Fellaini arriving from Standard Liege for £15million the following year.
Everton are now bracing themselves for an improved offer from United for the pair but Martinez is determined to keep them and insists there is no set price where they will begin talks.
He said: "No, no. Remember that the position we have got is we don't want to sell, so then you haven't got a figure.
"I don't think it is the time to talk about the window when you have just played a game and we have another game around the corner.
"I am not going to go along with the circus the transfer window can become."
Martinez certainly had no concerns about playing Baines and Fellaini against the Baggies.
Both performed well and Fellaini went the closest to scoring with a late shot that hit the post.
Martinez said: "They are true professionals. They are true ambassadors of Everton and they have been here for a long time and they know how to perform and give everything they have."
Everton goalkeeper Tim Howard also praised the professionalism of Baines and Fellaini and is keen for them to stay on Merseyside.
The American, himself a former United player, said: "It is entertaining. You look around at all the managers (saying) who is for sale, who is not for sale, then there is a fire sale at the end, all the big names.
"It entertains me who is going to Real Madrid, who is going to Chelsea, who is going to Man Utd.
"There is nothing we can do about it. What I will say is that Felli and Baines have been brilliant.
"Other clubs have got some idiots who don't turn up to training or train with the reserves. These guys (Fellaini and Baines) train so hard, they are out there working as hard as anyone else.
"What goes on in the boardroom is not down to me.
"Of course I want them to stay. They have been brilliant for us.
"Hopefully that will last a long time but guys a lot smarter than me are making those decisions, so I don't know."