> Lambda calculus doesn't have a concept of its own expressions being data manipulated by lambda calculus. There is no quote operator in lambda calculus.
McCarthy himself noted this about Lambda Calculus and came up with EVAL. In fact several people have also noticed this. They simply shrugged and delta-ruled all the things.
All the terms in lambda calculus are functions; there are no conses, symbols, numbers, strings, nothing. (Immutable) cons cells can be simulated in lambda calculus using functions. This being in the sense that functions resembling cons, car and cdr can be expressed such that car(cons(x, y)) = x, and cdr(cons(x, y)) = y.
Lisp arises from the realization that expressions built out of trees that consist of pairs can be executed using an eval function, in a way that incorporates some concepts from lambda calculus, such as anonymous functions.