No — shaving hair doesn't change its thickness, color or rate of growth.
Shaving facial or body hair gives the hair a blunt tip. The tip might feel coarse or "stubbly" for a time as it grows out. During this phase, the hair might be more noticeable and perhaps appear darker or thicker — but it's not.
If you notice a sudden increase in facial or body hair, talk to your doctor. This could be a medication side effect or a sign of an underlying medical condition. Your doctor might also provide advice about various hair-removal techniques.