No, BOTOX has absolutely no effect on thickening the skin. For that chemical peeling is helpful. Oddly, ablative laser resurfacing seems to have the opposite effect. However, laser does increase collagen in the upper dermis. The bad news is that laser can cause an atrophy in the more superficial skin which is why sometimes people who have had laser resurfacing look like the aliens have sucked their face off.
As we age, the skin gets thin. Botox has no effect at all on skin thickness. Botox softens the lines of facial expression but has no effect on the thickness or on the collagen in your skin.