This is a difficult question because to my knowledge from reading through the Bible multiple times there’s no examples of this in the Bible (if I’m missing any relevant passages please let me know in the comments). There are passages that I think are relevant:

*God has not hidden himself; we’ve closed our eyes to him*

Romans 1:18-23 (ESV; emphasis mine)

> 18For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who by their unrighteousness suppress the truth. **19For what can be known about God is plain to them, because God has shown it to them. 20For his invisible attributes, namely, his eternal power and divine nature, have been clearly perceived, ever since the creation of the world,g in the things that have been made. So they are without excuse.** 21For although they knew God, they did not honor him as God or give thanks to him, but they became futile in their thinking, and their foolish hearts were darkened. 22Claiming to be wise, they became fools, 23and exchanged the glory of the immortal God for images resembling mortal man and birds and animals and creeping things.

*The Biblical basis for salvation is faith in Jesus that comes from hearing the gospel (the message of how to be saved; it’s in many many places in the Bible)*

Romans 10:11-17 (ESV)

> 11For the Scripture says, “Everyone who believes in him will not be put to shame.” 12For there is no distinction between Jew and Greek; for the same Lord is Lord of all, bestowing his riches on all who call on him. 13For “everyone who calls on the name of the Lord will be saved.”
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> 14How then will they call on him in whom they have not believed? And how are they to believe in him of whom they have never heard?c And how are they to hear without someone preaching? 15And how are they to preach unless they are sent? As it is written, “How beautiful are the feet of those who preach the good news!” 16But they have not all obeyed the gospel. For Isaiah says, “Lord, who has believed what he has heard from us?” 17So faith comes from hearing, and hearing through the word of Christ.

*There’s a passage that addresses a form of asking God for something while doubting, and it’s not positive; faith is key*

James 1:5-8 (ESV)

> 5If any of you lacks wisdom, let him ask God, who gives generously to all without reproach, and it will be given him. 6But let him ask in faith, with no doubting, for the one who doubts is like a wave of the sea that is driven and tossed by the wind. 7For that person must not suppose that he will receive anything from the Lord; 8he is a double-minded man, unstable in all his ways.

*God doesn’t want anyone to go to hell, but all to be saved*

2 Peter 3:7-9

> 7But by the same word the heavens and earth that now exist are stored up for fire, being kept until the day of judgment and destruction of the ungodly.
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> 8But do not overlook this one fact, beloved, that with the Lord one day is as a thousand years, and a thousand years as one day. 9The Lord is not slow to fulfill his promise as some count slowness, but is patient toward you,a not wishing that any should perish, but that all should reach repentance.

**Bottom Line**

While God often has mercy on us in our weakness and sinfulness, based on the lack of positive examples in the he Bible of such a prayer and based on the above passages, I do not see a reasonable expectation that God would answer such a prayer because 1) he has already given us all the evidence in Creation we need of his existence, 2) he has established things so that we believe by hearing the gospel—he wants us to found our faith in his word, and 3) he desires us to put our faith in him (that he exists), but more than that in his Son Jesus and be saved. Faith is **the** key of Christianity and of the gospel—it is how you are saved and it is what separates those going to hell from those going to heaven. Praying and asking God if he exists runs counter to all this. If God answers such a prayer then it is in his mercy, and he is a merciful God! But I see no reason to pray such a prayer when God has established a biblical way to learn about himself.

If you want to know if Christianity is true, read the Bible. Talk to pastors and other mature Christians. Find out about the gospel. And of course if you want to know if God exists open the eyes of your heart as you look around you at nature. There’s no need nor any biblical basis of which I’m aware for putting God to the test for a personal sign to see if he exists and Jesus is the Messiah. 

In other discussions about this topic on this site I’ve seen passages used in support of seeking a sign that mention signs in the context of faith but those signs are freely *given* by God through the apostles; when demanded, e.g. by those of Jesus’ fellow Israelites who rejected him, it was rebuffed by Jesus himself (look up “the sign of Jonah” in the Bible). And other passages about God asking the Jews to test him in his promises by tithing as they were required by the Law are out of context of this discussion, and arguably different—there God is basically begging them to believe him and his promise, that they should willingly give up some of their livelihood, a crazy thing in human wisdom, because God has promised to bless them over and above what they gave up (the tenth or tithe). Anyone arguing that we are to put God to the test in the context of this question needs to make sure to use such passages in context, and from what I see in the Bible, they do not support the idea of such a seeker’s prayer being Biblical. Seeking is good (there is no better thing than to find Jesus) but it must be on God’s terms, not ours.