No, that's not it at all.

The p sound didn't change, either in Europe or in China. The Wade-Giles romanization system recognized that Mandarin Chinese doesn't distinguished voiced and unvoiced consonants like European languages, but rather aspirated and unaspirated. So p' (with an apostrophe) was used for an aspirated p sound and a plain p was used for an unaspirated one.

Unfortunately, these apostrophes were frequently omitted by people uninterested in accurate Chinese pronunciation, so one often saw things such as "Tang dynasty" instead of "T'ang dynasty". For that reason among others, a new romanization system called Pinyin was developed. Now an aspirated t was written t instead of t' (incidentally making "Tang dynasty" actually correct) and an unaspirated one was written d. Likewise, an unaspirated p was now written b, and so that's why Beijing starts with a b.

As for king becoming jing, that was a Chinese pronunciation shift, not a European one. The letter k has not changed in pronunciation much in over 2000 years.