There have been suggestions that the Scotsman, who spent 11 years on Merseyside, has tainted his relationship with the Toffees fans after poaching midfielder Marouane Fellaini in the summer.
Moyes was also in pursuit of full-back Leighton Baines, but he failed to clinch his signature by the end of the transfer window.
According to ITV Sport, when asked if Moyes has tainted his relationship with the fans, Martinez said: "I don't think that is something that I spend much time [thinking] about.
"I think David Moyes did a fantastic job over the last 11 years and that is what I want to keep.
"Everything else that has happened are normal situations which would occur in football."
Moyes will come up against his former club at Old Trafford on Wednesday.