Born in Aarhus, Gertsen started out in the youth team of AGF before joining Silkeborg IF in 2009, initially with the under-13 team.[1][2] He gradually ascended up the ranks during his teenage years, while simultaneously studying at Silkeborg Gymnasium, which he completed in 2017.[1] In the summer of 2015, while a part of the under-19 squad, his contract was extended for three more years. It also was around this time that he started training with the first team on occasion.[3] He appeared on the substitution bench for the senior squad a total of eight times during the 2015–16 and 2016–17 seasons.[4]
Gertsen made his first-team debut on 19 September 2017, playing the full 90 minutes of a second-round Danish Cup match against Kjellerup.[5][6][7]
He made his Superliga debut on 10 December 2017, replacing Davit Skhirtladze in the final minutes of a 2–1 victory over Odense.[8][9] The next month, Silkeborg signed him to a three-year contract extension, securing him until the summer of 2021.[9][10]
On 28 January 2021, Gertsen signed a pre-contract with FC Fredericia, valid from 1 July 2021, where his contract with Silkeborg also expired.[11] On 16 January 2024 FC Fredericia confirmed that Geertsen had left the club by mutual agreement, a year and a half early.[12]
On 3 February 2024, Gertsen joined Icelandic club Vestri.[13] He left the club again at the end of the year.[14]
On March 19, 2025, Gertsen joined Danish 2nd Division side Næstved Boldklub.[15] At the end of June, Gertsen moved to fellow league club AB on a deal until June 2028.[16]