生事 is somewhat formal/literary. Everyday alternatives: 闹事 (nàoshì, 'cause a disturbance', stronger and more common), 惹麻烦 (rě máfan, 'stir up trouble', colloquial). Common collocation: 无事生非 (wúshìshēngfēi, idiom: 'making trouble out of nothing').
memory
生 'to produce / give rise to' + 事 'matter / affair' → literally 'produce an incident' = make trouble. The 生 here is the same productive 生 in 生病 'to fall ill' and 生气 'to get angry' — 'produce' an unwanted state.
