No. The act of transferring a balance to a new credit card in and of itself does no damage to your credit score.
However, there are two things that might negatively affect your credit worthiness about the process if you're not careful.
Firstly, applying for a new credit card is always recorded on your credit history. That means applying for a lot of cards in a short space of time can make other lenders nervous about offering you a product as well.
Card finder and eligibility checker tools don't show up to other lenders on your credit score, however, so it's wise to use one before applying to limit the number of applications you'll have to make.
Secondly, lenders look at something called your "total available credit" before making a decision. This is worked out by adding up all your credit card and overdraft limits.
A new card adds to this total, but unless you've already got a large number of cards with high credit limits - for example equal to your annual salary - it probably won't be a problem.