HI Cristina The second one does not exist in English (you cannot use it this way). Possible is an adjective. Possibly is an adverb. So the full sentence would be "It is (or was) quite possibly the best season," in which the adverb possibly provides more information about the verb "to be." Possible should be used with a noun - for example "He is a possible suspect in the investigation."

The word possible doesn’t seem to fit this context.