Literally: peace – calm / quiet – no – thing
Alternately: Peace and quiet; safety and calm. Uneventful, in a good way. This phrase evokes the pleasant side of mundane day-to-day life, on both the personal and the societal levels.
Notes: This is a compound of compounds; 安穏 is “peace,” “tranquility,” while 無事 is literally “uneventful,” by extension “peace,” “safety.” There is a wide variety of synonymous phrases, many of which incorporate 無事, such as 無事安穏 (this compound but with its halves reversed) or 平穏無事 (heion buji).
Note that the onyomi for 穏 is normally on and in 安穏 it can be read that way (i.e. an’on), but due to phonetic slurring annon has become standard.