Chronologically the two met as young kids after the Pilaf gang wished for youth to Shenron just seconds before Future Piccolo's death. Both Future Trunks and Future Mai momentarily looked at one-another
before the latter departed. Sometime prior to the "Future" Trunks Saga, Trunks and Mai met and became allies that stood in defiance to Goku Black.
Trunks is quite close to Mai and has saved her life on a number of occasions (and vice-versa). When he thought that Goku Black had killed her, Trunks became so enraged that he immediately attacks the dark entity. After returning to his time, Trunks was tearfully relieved when he found out that she was still alive. After returning to the future the second time and learning that Mai had gotten severely injured from fighting off Black, he takes action and heals her with a senzu bean in a kissing fashion. After the deaths of the remaining Earthlings from Infinite Zamasu's assault, Trunks consoles the grief-stricken Mai.
Humorously, whenever other characters mention the possibility of them being in a relationship, both Trunks and Mai deny it while blushing at the same time