What is the adverb of the verb relax?

It depends on whether you want the adverb to relate to 'relaxing', 'relaxed', or 'relaxative'. 'Relaxingly' is a possibility, though it's not really idiomatic. I think it would be better to say 'in a relaxing manner', 'in a relaxed way', or a similar construction, using whichever adjectival form you want to refer to.