The phrase "inconvenient moments" refers to times when something happens that is not convenient or suitable, making things difficult or uncomfortable. Full definition
Moms are busy and babies always want to be held at the most inconvenient moments, those are just simple facts of life. (rookiemoms.com)
Our greatest, most life altering problems always seem to reveal themselves in the most mysterious of ways at exceedingly inconvenient moments. (counselingcenter.org)
Her family suffers too, although they have gotten used to the other people in the house, who turn up at the most inconvenient moments. (newsday.com)