The former Chelsea star has just one goal and two assists in 13 games this season since his big-money move from Stamford Bridge in the off-season.
Hazard began the campaign by missing the first month after a hamstring injury that followed him arriving to his new club overweight.
Still, Martinez believes the Belgium playmaker will become a Madrid star.
"I saw Eden at the [Santiago] Bernabeu against PSG and I have to say that we saw a fascinating hour," Martinez said on Saturday.
"He taught us how his season with Real Madrid will be and the truth is that it makes your mouth water.
"Too bad he had to come off because of the injury, but we see an Eden Hazard that is going to contribute more and more, that his mates know him more and more and the truth is that the project of Eden at Real Madrid will be very special."
Real Madrid have confirmed that Hazard's injury is only a bruise, dispelling fears that he could be set for a lengthy period on the sidelines.
Still, Madrid would not provide a time frame for the Belgian's recovery, and he missed out on Saturday's 2-1 win at Alaves.
Another player sure to be key to possible success for Belgium at next year's tournament is Romelu Lukaku, who has scored 11 goals in 17 games for Inter this season.
"The truth is that he had a very difficult period in Manchester United, especially the last six months," Martinez said about the striker.
"He is now leading a project with Antonio Conte and it will be very interesting to see its evolution."