In the absence of agreement to the contrary, in the law of England and Wales, a contract is formed when:

• There is a meeting of minds – both parties agree on the terms of the contract, one offers, and the other accepts
• Both parties intend to be bound contractually
• Both parties give consideration for the contract

By "agreement to the contrary" I mean for example a term stating that "no contract is formed until goods are dispatched" as is common in online retail.

In this case, the question is whether the contract is concluded on the telephone (the agent offers and the passenger accepts), or is concluded subsequently (the agent quotes a price, the passenger offers to accept it, and the acceptance by the airline occurs when they ticket it). Both are potential interpretations and I am not sure what a court would rule.

The question of consideration does not arise here in that a promise to pay can constitute good consideration.

I see the passenger eventually did get their way which is good.