The Paraguayan defender followed Martinez to Goodison Park from Wigan after his contract expired at the end of last season, and the Spaniard was impressed with his first outing for the club.
And after watching his first outing, Martinez expects the defender to become a valuable asset at Goodison Park.
"It's important he got that first game and got to know his team-mates," the boss told the club's official website."
"It feels like he has been with us for a long, long time and he plays with that maturity.
"That is down to the group of players that we have who have welcomed the new signings really well.
"It's pleasing to see Antolin settle in really quickly. That's always a nice sign of a dressing room."
Martinez brought the 30-year-old to England in 2010 to tighten up his Wigan defence, he did just that as the club survived relegation from the Premier League in two consecutive seasons.
However, injury problems restricted him to just 77 appearances during his time at the DW Stadium.
Alcaraz struggled with a groin injury during the last campaign, playing just 14 times, but showed no signs of this issue after coming on against the League Two outfit.