"언니랑 성욱 씨랑 관절생일에 작은 선물 하기로 했어요" - WRONG
(I decide to give a gift to my sister and Sung Wook for their combination birthday)
관절생일 doesn't exist - no one says 관절 and 생일 together as a phrase.
Probably it was originally written as "joint birthday", joint meaning "combined".
But joint also means the articulations(knee joint, elbow joint) in the body, and the Korean term for that is 관절 (무릎 관절, 팔꿈치 관절). 관절 however is only used for the body joints, never for general "combination", so 관절 생일 is just comical.
For "joint birthday", "합동 생일" or "합친 생일" is probably the best term.
합동 is a noun and adjective meaning "joint, combined, merged", although a bit formal ("joint concert" == 합동 공연).
합하다 and 합치다 are the corresponding verbs: "join, merge".
So the sentence may be translated as:
=> 언니랑 성욱 씨(의) 합동 생일에 작은 선물(을) 하기로 했어요.
=> 언니와 성욱 씨 합친 생일날에 조그만 선물(을) 하기로 했어요.
=> 언니랑 성욱 씨 생일을 합쳐서 선물을 하나 준비하기로 했어요.