The answer is nuanced, but essentially, yes, "I Am" is related to God's name.
According to Exodus 3:14, God reveals his name to Moses as אֶהְיֶה אֲשֶׁר אֶהְיֶה, Ehyeh Asher Ehyeh. This is most basically translated as "I am that I am" or "I shall be what I am". The term "Ehyeh", which translates to "I Am," is directly connected to God's revealed personal name. Therefore, "I Am" is an aspect of God's self-revelation and integral to understanding His nature and name in the Hebrew Bible. While not the only name, it is a significant and defining aspect of God's identity revealed to Moses.