A lot depends on how you or your business understands time.
Quarters are based on months, and in the Gregorian system months are solar based so they're independent of weeks. In most cultures, the quarter is based on a mix of astronomical and legislative factors. This means that there can be small differences in when a quarter starts/ends depending on the country being considered.
If you use one of the lunar calendars (Hebrew, Arab, several Asian), then the rules for determining the calendar Quarter are quite different. I won't go down that particular rat hole unless it is germane to your question.
Weeks are another interesting point. The clerical week always begins on the Sabbath, but the business week typically begins on the day after the Sabbath. This gets "interesting" in countries that have multiple Sabbaths.
Because of this, we need to understand how you understand/define time and also what exactly you mean by "the start of the week of the current quarter" in order to answer your question.
In theory, theory and practice are identical. In practice, theory and practice are unrelated.