Although you can claim and examine the egg at level 1, you must be level 86 to complete the quest series that starts with Border Defense in Goahmari Village. That quest series gains you enough faction to build the trust of the Goahmari gryphon expert Kezu, who will teach you more about your egg and its care.
I should consult a griffon trainer in the Village of Thundermist
You are looking for <senior gryphon trainer> Elize Flannagan at ( 623, 4, -234 ) /waypoint 623, 4, -234 in The Thundering Steppes. This is the Thundermist Village griffon tower station, just outside the village.
I should consult a goahmari gryphon expert
Speak with <Gryphon Expert> Kezu at ( 366, -14, 477 ) /waypoint 366, -14, 477 in Goahmari Village in the Great Divide. He will take the egg away from your for safe keeping.
I should return to gryphon expert once I have proived I am a trustworthy guardian for a young gryphon.
At this point, you must complete either the adventurer or tradeskilling quest series to learn how to raise a gryphon and earn the trust of Kezu. (L86+ is needed to start either)