No. This was tested as far back as 1928 and again in the 1970s, with the same result each time. A razor cuts the strand at the surface and never touches the follicle, which is the only thing that decides thickness.
Transcript
Shaving makes it grow back thicker. Every teenager believes it. It was tested in 1928 and again in the seventies. Same result. A razor cuts the strand at the surface and never touches the follicle, which is the only thing that decides thickness. What you're seeing is the blunt tip, a hair that tapers naturally looks fine. Cut it flat and the same hair looks coarse.
What you are seeing is the blunt tip. A hair that tapers naturally to a fine point feels soft; cut it flat and the same hair feels coarse. Nothing about the hair changed except its end.