Mathias Greve was born on 11 February 1995 in Langeskov, Kerteminde Municipality, on the eastern part of the island of Funen, Denmark.

During one of his initial training sessions at local club Langeskov IF, an incident where he slipped in a puddle left him disinterested in the sport for the next couple of days.

[3] Sports ran deep in Greve's family, and his relatives played football, handball, and badminton which shaped their daily lives.

[3] Greve quickly grew into a talented footballer, and during school recesses he was recognised as someone who could easily dribble past four or five classmates, displaying a finesse that set him apart.

After taking a gap year following high school, I attempted to study law, but it proved to be quite challenging alongside football.

However, his eventual growth surge rekindled his talent, as he was able to become more dominant physically, and during his under-19 years, both Greve and OB noticed significant development.

This progress led to an offer of a part-time contract that swiftly transitioned to a professional one in April 2014 due to his ability to excel among more experienced players.

[9] Upon extending, he was praised by head coach Ove Pedersen: "He has demonstrated a fantastic attitude, and it's even more impressive as he has delivered results both in practice and matches, even when he has occasionally been sent down to the under-19 squad.

[13][14] Greve started the 2015–16 season as a substitute, as players such as Mikkel Desler and Kenneth Zohore were preferred in his position under new head coach Kent Nielsen.

[1] The limited playing time continued into the 2019–20 season, and in January 2020 he stayed home from OB's training camp in Belek, Turkey, to negotiate with other clubs.

[22][23] On 23 February, he made his debut for the club in a 0–0 home draw against his former side OB, replacing Tosin Kehinde in the 66th minute.

[31][1] After a strong start to the 2021–22 season with two assist and one goal in the first three games of the campaign for Randers, Greve was sold to 2020–21 Danish Superliga champions Brøndby on 4 August 2021, signing a deal until June 2025.

He started on the bench but came on in the 52nd minute for Mathias Hebo in a 1–0 victory over the United Arab Emirates U20 at Slovenský Grob, Slovakia.