“But if you say, “show me thy God,” I would reply, “Show me yourself and I will show you my God.” Show then, that the eyes of your soul are capable of seeing, and the ears of your heart able to hear.” St. Theophilus to Autolycus, 1.2
Proverbs 26:4 warns us not to “answer a fool according to his folly.” This means that we must be careful when responding to unbelievers, that the context must be appropriate. If they are being testy and do not really desire to know God then we should not respond as if they did, otherwise you set your own trap to fall in. They are asking you to “show” God through pure logic, which simply cannot be done. There are elements of faith and belief that must be present, even in the smallest form of a mustard seed, as Christ mentions in the Gospel. You can tell if someone has this small amount of faith merely by the context of the conversation or even posture of their voice.
God is not going to force his way into the soul of the unbeliever. Their must be some pain somewhere within this unbeliever for them to be interested in a relationship with God. When I say pain, I mean that there must be an amount of humility within the soul that allows such a conversation about God. First Peter 5:5 says that God gives grace to the humble but resists the proud. If the unbeliever could simply find just an ounce of pain in their life that they are willing to table, willing to show you, that is when the time is right; not that they will spill their guts to you; The pain itself may remain hidden in their mind but it may manifest to you in the form of listening to what you have to say about the Living God, thus “showing” themselves to you!