This is most likely Huntik, an American-Italian cartoon aired between 2009 and 2012.
Two teenagers and two adults, there was a teenage girl and an adult woman who used to be a spy before working with them.
Actually one adult (Dante, the redhead one) and three teens: Lok, the blonde guy, Sophie, the skirt girl, and Zhalia, the dark-haired girl on the right. Zhalia was raised by a member of the Organization (the bad guys' team) and acted as a spy for them, even betraying the crew she had come to befriend, but later rejoined them in episode 20, The Unseen Guide.
They're summoning differents monsters.
This is literally everything the show is about, as the four main characters are "Seekers"; Seekers can summon creatures known as the Titans with the help of several amulets. Here's a sheet with all Titans and their summoning amulets.
In the first episode, the girl met a guy in high school and wanted to copy his notes.
It's the other way around: Lok (blonde guy) asks his classmate Sophie (skirt girl) to help him study. In fact, he challenges her to do so if he manages to beat her crosswords in less than two minutes. He succeeds, so as a deal is a deal, Sophie goes to his place in order to help him with school subjects. Series of events happen, they find Lok's father's journal, discover he was a Seeker, and the adventure begins.
Later, the four are questing for legendary monsters in Atlantis, which has five temples.
That's season 1, episode 21, Coming of Age. The team travels to Atlantis after having escaped from the bad guys in the previous episode. There, they have to prove their worth through four tests held in temples of Poseidon (there are five temples of Poseidon in the city, and a ruined one): bravery, prowess, intelligence, and nobility. they encounter the local Titans.
Below is said episode (in French, though):
Found by habit, because for whatever reason I keep identifying this show, as per my answers to: