When any question is asked like this, there are some important things to remember. First, we can only know that God does or does not do something based on what Scripture tells us and not on human conjecture. Second, acceptance of a Bible truth does not force the acceptance of a falsely devised premise based on that truth (example: it is true that Lydia’s household was baptized, but that does not force one to accept infant baptism). Third, the context of a passage must be carefully weighed (context includes the speaker, the setting, the verses surrounding, and remembering to keep it in harmony with the whole of Scripture).
We need to be careful drawing conclusions and basing arguments on John 9:31. The statement in that verse (“we know that God does not hear sinners”) was not made by an inspired man but was a response made by a formerly-blind man to some dogmatic Pharisees. That doesn’t automatically deny the truthfulness of the statement, but it also doesn’t automatically make it true. The matter would need to be affirmed elsewhere to make it so.
There is the example of Cornelius in Acts 10. He definitely was not in a saved condition (see Acts 11:14) when he “prayed to God always” (10:2), but an angel came and told him, “Your prayers…have come up…before God” (10:4). God heard this non-Christian’s prayers, but was it just an exception for the whole purpose of taking the gospel to the Gentiles? It is important to note that irrespective of God hearing his prayers, Cornelius was definitely not saved by prayer.
Jesus taught, “If anyone wants to do His will, he shall know concerning the doctrine” (John 7:17), and, “Blessed are those who hunger and thirst for righteousness, for they shall be filled” (Matt. 5:6). If a sinner prays, desiring to find and know God’s will, what commitment has the Lord made to such a heart in these passages?
The law of Christ is amenable to all, therefore, these words in Matthew 12:36-37 apply to all: “Every idle word men may speak, they will give account of it in the day of judgment. For by your words you will be justified, and by your words you will be condemned.” Would this include words that a non-Christian might use in a “prayer”? Does God hear those words?
The argument cannot be sustained that God does not hear an alien sinner’s prayer, as the whole of Scripture does not support such. However, that in NO way lends credence to contriving or accepting a falsely devised premise that God would save an alien sinner by means of a prayer. God’s clearly revealed plan of salvation will not ever justify that notion.