Properly refer to the turn of the year

During a meeting I was explaining a problem that only occurs once year: when one year ends and new one begins. Specifically during the first few days of the new year. Unfortunately, I was lost for words to describe the phenomenon, and, in the midst of stuttering, uttered, "Well the problem only happens during year turning, umm, well when 2010 became 2011..umm.." I got my point across, with mild sense of embarrassment, considering that a fairly large audience was attentively listening to my poorly formed narrative (I'd better stop now with this extraneous information).

What is the best way to describe what I was trying to telegraph --that is, the issue only occurring during the turn of the year?

The phrase in your question is good:

The issue only occurs at the turn of the year

(if it only happens at the moment the year changes), or perhaps

The issue only occurs around the turn of the year

(if it happens for a longer time period)

Or you can paraphrase:

The issue only happens when the year changes from 2010 to 2011

(though that might imply it only happened in those specific years), or

The issue only happens in the first few (hours, days) of January

You could use the term year-end, which refers to the period at the end of the year, as in year-end awards, year-end sales, year-end charitable giving, and in your case, a year-end problem, e.g. a problem that occurs at year-end.