So continuing to use your carrot example, If the merchant usually bought 100 pretax carrots how many fewer post-tax carrots will the merchant buy from the farmer?

Don't know. It's an empirical question. We just know it will be fewer.

The merchant knows that he will be able to sell fewer carrots after the imposition of the tax, because customers will buy fewer carrots. Demand curves slope downward - if you raise the price of a good, people will buy fewer than if the price were lower.

There are some theoretical and perhaps a few real-world goods that don't exhibit that (OMG Giffen goods), but nearly any good like a carrot will see quantity demanded fall in response to an increase in price. So you won't see the farmer pass along 100% of the tax to the merchant, and you won't see the merchant pass 100% of his increased cost on to the customer.

Beyond knowing the direction of the price and output movements (and being very confident that neither will exhibit 100% price inelasticity), we don't know the amounts of those movements. That will depend on how sensitive the farmer, merchant, and customer are to price changes.