Only if you define the birth of smartphones as starting with Android/iPhone. Palm released their own smartphones using the same OS as their PDAs (the Treo line) prior to that, and Microsoft had Windows Mobile plus a bundle of third-party manufacturers. Plus Blackberry had been around the whole time, too.
Looking it up, it seems that Palm Tungsten TX[1] was produced in 2005, 1.5-2 years after they started producing the Treo 600[2], which was one of the first smartphones - yes, it was inferior to the iPhone in a number of ways, but having owned a Treo 680, in my mind it still qualifies as a smartphone.