In conversational English, we usually round our time up or down to the closest quarter hour. When it's "just after" 2:45am, it is like the time that is approximately in between 2:45 - 2:50am. If the time gets closer to 3am, we would say it's "just about" or "almost" 3am.

'Just after' means 'shortly after' or 'just a short time after'. 'Just after quarter to three' might mean 2.45 am and a few seconds, if you're being precise. Or if you're being less precise, it might be 2.46 am or even slightly later.

It means a short time after

1-2 minutes after quarter to three. It's like a bit after/only a bit after.

I think that is still three, or that is still very early - one can argue if 3 am is morning.