In this paper, we utilize Tribonacci numbers, which are a generalization of Fibonacci numbers. Let T_n be the n-th Tribonacci numbers defined byT_(n+3)= T_n+T_(n+1)+T_(n+2); T_0=0,T_1=T_2=1. Here, we introduce a novel concept called Tribonacci Heinz Quarter Mean labeling. An injective function f:V(G) → W is said to be Tribonacci Heinz Quarter Mean labeling if the induced function f ∶ E(G)→{T_1 〖,T〗_2,…………,T_r } definedby(????=????????)=⌊(∜(f(u)f(v) ) (√(f(u))+√(f(v))))/2⌋or⌈(∜(f(u)f(v) ) (√(f(u))+√(f(v))))/2⌉ ,then the resulting edge labels are distinct is called Tribonacci Heinz quarter mean labeling and a graph which admits Tribonacci Heinz quarter mean labeling is called Tribonacci Heinz quarter mean graphs.