No. The act of transferring a balance to a new credit card doesn’t damage your credit score.
However, there are two things that might negatively affect your credit worthiness if you're not careful.
Firstly, applying for a new credit card is always recorded on your credit history. This means that if you have an application rejected, or there are multiple applications over a short period of time, other lenders might be nervous.
However, card finder and eligibility checker tools don’t show up on your credit score, so it makes sense to use one to ensure you only apply for cards you’ll qualify for.
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’s unlikely to be a problem.