My understanding is that in order to be considered to have commited a Chilul Hashem, the desecrating act must have been committed in front of a certain minimum quorum (e.g., a Minyan). Is my understanding correct and if so, what is the minimum number?

A Chilul Hashem can be done even in front of one person. However, if it was done in public- in front of Ten people, a "Minyan", the Sin is a lot worse. For example: someone who desecrated the Shabbat in public is considered a gentile for certain Halachot. Rabbi Yitzchak Grinblat