According to this lecture by Prof. Wildberger on the history of mathematics, and the introduction of projective geometry..
Projective geometry began with the work of Pappus, but was developed primarily by Desargues, with an important contribution by Pascal.
Although if you are asking about the word and not the work, I don't know.
I looked in MathWords and found this
PROJECTIVE GEOMETRY. In his account of the origins of projective geometry Katz (1993) mentions Pascal, Desargues and Monge before saying that J.-V.Poncelet composed the first text in synthetic projective geometry: his Traité des Propriétés Projectives of 1822. See here for an extract from Poncelet’s Introduction.