Ioakeim’s name day is September 9, the feast of Joachim and Anne.
Churches that follow the Old (Julian) Calendar keep the same feast on September 22 of the civil calendar. See everything commemorated on September 9.
The saint behind the name

Joachim and Anne
September 9, Saint, Ιωακείμ και Άννα
Saints Joachim and Anne were the parents of the Most Holy Theotokos. Righteous and devout, they remained childless into old age, bearing reproach with patience and beseeching God with tears; in answer to their prayer they were granted a daughter, Mary, whom they dedicated to the Lord in His Temple.
Also celebrating on September 9
These names share the feast, many of them other forms of the same name:
What to say on a name day
The Greek wish is Χρόνια Πολλά (Chronia Polla), “many years.” A call or a short message on the day is what people remember. See name day wishes you can send and what a name day is.