Larry Hurtado writes:

In general usage, a “conversion” marks a change from one religion to another, or a shift from an irreligious to religious profession/stance.  At the time of Paul’s experience (a scant couple of years after Jesus’ crucifixion), the Jesus-movement wasn’t what we know and think of as a self-standing “religion.”  It was more a rather exclusive new sect or movement within the larger Jewish tradition.  (And it must be emphasized that Paul’s “persecution” of Jesus-followers was not directed at “Christians” but solely at fellow Jews whom he must have regarded as having seriously problematic in their beliefs and practices.)

More significantly, Paul refers to that experience that prompted his shift in direction as a “revelation” (apokalypsis) and a “calling” (kaleo) as in Galatians 1:11-17.  On the other hand, Paul can refer to those Gentiles who accepted his gospel message as having “converted” or “turned” (epistrepho) to God and having turned away from their ancestral gods (“idols”), as in 1 Thessalonians 1:9-10.  So, in Paul’s thinking Gentiles/pagans “convert” from their polytheistic practice to worship and serve “a true and living God.”  But Jews such as he instead come to right understanding of what their ancestral deity requires of them.