This can only be answered from a literalist Young-Earth Creationist (YEC) perspective, because to all other perspectives it's a moot point, so I'll answer from that viewpoint.
I might as well throw in my usual disclaimer that I'm not debating whether the YEC view is true, this is just explaining the YEC view. Such debates in comments are off-topic and not constructive per site guidelines, and not relevant to the question/answe at hand. Even though I personally believe in the YEC view, I'm not defending it here, just explaining it.
Given the definition of time, taken from Mirriam-Webster:
b : a nonspatial continuum
that is measured in terms of events which succeed one another from
past through present to future
The very first event was in Genesis 1:1 (KJV)
In the beginning God created the heavens and the earth.
This was the first event. The universe didn't exist at all before this point. At this point, the universe began, thus, at this point, the continuum of events began. By definition, this is when time began.
Genesis 3 would mark the first normal, regular, repeatable events by which could measure time began, but time actually began with space and energy in Genesis 1:1, when God spoke the universe into existence.
This is similar to the answer given on the Answers in Genesis site - a Young-Earth Creation and apologetics organization:
Although the argument is increasingly common, nowhere in the text does
it say that time began on the fourth day. Instead, God made the sun,
moon, and stars, which can be used to measure time. Time actually
began "in the beginning" (Genesis 1:1), or else it would not have been
"the beginning," which is a time reference. In fact, "day" is a time
reference, and there are three of those prior to day four.
Revision of the first sentence:
This answer would also hold true for proponents of the gap theory, day-age theory, and others. The beginning is the beginning regardless of how long it's been since then.
It should probably be changed to "This can only be answered from tho position that you believe the Genesis account to be true, because to those that don't it's a moot point."