I don't fully understand the phrasing of the question but I'm going to guess that
The second hint is interpreted literally so that "Month/Day = A quarter = 1/4" so her birthday is the 4th of January. Notice this question was posted on the 5th but also the OP refers to the question as if it were first asked in the past so I think this is plausible.
For the other part
Quarter of a perfect year puts me in mind of quarter of a year that is a perfect number. The only perfect number that would really fit our timeframe is 8128 and one quarter of that is 2032 so I think really that is the year that Aica gets excited about
So the full answer is probably
1/4/2032 - although her birthday falls on the 4th of January every year, in any case.