To unlock the Burnout Apocalypse trophy in Days Gone, you will first need to progress through the game until you reach the Diamon Lake camp and reach trust level three by completing story missions, encampments, clearing hordes etc so that you can purchase the Nitrous 3 upgrade.
Once you have the Nitrous 3 upgrade, you will be able to use the nitro boost for over the five seconds requires. Nitrous 2 only lasts for 4.5 seconds, so won't be enough to unlock the trophy.
To perform the drift, find an open space so that you can drift in a large circle, and press the left stick in the direction you want to drift, whilst also pressing
,
and
.