Did Trump win either of the Nobel Peace Prizes he was nominated for?

No he has never won any Nobel Peace Prize.  Abiy Ahmed won it in 2019, Nadia Murad and  Denis Mukwege won it it 2018, In 2017 it was won by the organisation the International Campaign to Abolish Nuclear Weapons and in 2016 it was won by Juan Manuel Santos

For a nomination alone, the barrier to entry is low: all nominations from heads of state or politicians serving at a national level are accepted.

University professors, directors of foreign policy institutes, past recipients of a Nobel Prize and members of the Norwegian Nobel Committee are also among those deemed qualified to submit a nomination for the prize. 

For the 2020 Nobel Peace Prize - the winner of which has not yet been announced - there were 318 candidates. 

He received 3 nominations but did not win since the Nobel committee is left wing.