Are you dealing with calendar quarters or relative quarters? Meaning do you want all quarters to start Jan 1, Apr 1, Jul 1, and Oct 1?
If you want calendar quarters, try:
DATETIME_DIFF(
DATETIME_PARSE(DATETIME_FORMAT(TODAY(), "YYQ"), "YYQ"),
DATETIME_PARSE(DATETIME_FORMAT({Start Date}, "YYQ"), "YYQ"),
"Q"
) + 1
If not, it sounds like you want the number of months between the start date and today, divided by 3:
IF(
{Start Date},
DATETIME_DIFF(TODAY(), {Start Date}, "months") / 3 + 1
)