I do not remember ever having seen or heard the word, 볼멘소리, until I read it in this article. Here is the relevant part of the article, which, by the way, has at least one grammatical problem:
통일부는 9일에는 근로자 격려차 개성공단을 참관한 정 장관의 남북출입사무소(경기도 파주시) 현지 기자회견 구상을 밝혔다가 반발을 샀다. 기자들 사이에서는 "인권대회 취재를 막거나 관심을 돌리려는 것 아니냐"는 볼멘소리까지 나왔다.
On the 9th, the Ministry of Unification received a backlash. While [an official] was explaining plans for a press conference at the North-South Entry Point Office in Paju, Gyeonggi Province with Minister Jeong, who had visited the Gaeseong Complex to give a pep talk to workers there, someone among the group of reporters asked in a sullen tone, "Were you trying to block coverage of the civil rights conference or divert attention from it?"
볼멘소리 is a compound word created from the pure-Korean verb 볼메다, which means "have an angry attitude" (성난 태도가 있다), and 소리, which means "noise." The current Korean definition of the word is "성이 나서 퉁명스럽게 하는 말투," but the original meaning of the word was "볼이 메어질 정도로 부어서 하는 소리," which seems to mean, "an angry noise made after filling up one's cheeks (with air)." Actually, I am only assuming that 볼이 메어지다 means "fill one's cheeks with air." If it does, then that would imply 볼멘소리 is "a statement or question 'blurted out' in an angry tone."
By the way, the first sentence from the Korean article is grammatically wrong because it has two topics (i.e. 통일부는 and 9일에는), which is a no-no. A sentence should have only one topic, so the "는" in 9일에는 should be dropped.