Question 1. is, strictly speaking, ungrammatical, and vague in most interpretations. If by *graph algebra* you mean *algebraic graph theory*,
then the question is hopelessly broad. If by question 1. you mean
``What is the relationship between *graph algebras* (in the sense of Lovász) and *quantum graphs* (also in the sense of Lovász),
then the question is extremely specific and easy to answer: a quantum
graph (in the sense of Lovász) is an *element* of a graph algebra
(in the sense of Lovász). A definition of both these notions
is given in e.g.

Hatami, H. and Norine, S.: Undecidability of linear inequalities in graph homomorphism densities. Journal of the American Mathematical Society, Vol. 24, No. 2 (2011), p.547--565

which would also be my answer to Question 2.
This is a good article to start,
rich in mathematical context
(e.g. Artin's solution of Hilbert's seventeenth problem).