My opponent just conquered battlefield “hall of legend”, my opponent choose not to pay to ready his legend. Is there any window within this interaction for me to play reaction card since my opponent choose not to pay for the battlefield effect?

No — once your opponent declines the [1], there's nothing left on the chain for you to respond to.

Hall of Legends' ability finalizes like any trigger, and "you may pay [1] to ready your legend" is a cost (it has a linked effect). The opponent pays it, or declines to pay it, during the Pay Costs step of finalization, before the item ever becomes a Finalized Chain Item — if they decline, the ability is removed from the chain at that point and the game never enters a Closed State (CR 405.2, and FAQ #11075 for the timing). So there's no priority window attached to the declined trigger; you can't punish the choice with a Reaction.

The flip side: if they do pay, the ability does become a Finalized Chain Item and you get priority to respond to it. Also note the trigger is fully public information — they must make the decision as the trigger is placed, not later during resolution, so you'll know the outcome before anything else happens.