To the OP question the answer IMO is not whether they worship the same God but rather do they accept the offering of salvation "Jesus" If they don't I assure you they don't know God the "I AM ".
Something else, I believe that you must have a proper understanding of Jesus and salvation to be born again of the Spirit.
Paul said in Romans
Romans 12:1 I beseech you therefore, brethren, by the mercies of God, that you present your bodies a living sacrifice, holy, acceptable to God, which is your reasonable service.
2. And do not be conformed to this world, but be transformed by the renewing of your mind, that you may prove what is that good and acceptable and perfect will of God.
He here is talking about presenting your bodies and minds to God. Paul said be transformed by the renewing of your mind. James said receive with meekness the engrafted Word which is able to save your soul. The engrafted word will renew your mind and save your soul. They both said a similar thing about renewing restoring and saving the mind or soul.
The 23 Psalm said a similar thing... Psalms 23:3 He restores my soul; He leads me in the paths of righteousness For His name's sake.
He restores my soul.
Paul then reflects on this renewed mind, speaking of himself in Romans chapter 7. The last verse says.
[sup]24[/sup] O wretched man that I am! Who will deliver me from this body of death? [sup]25[/sup] I thank God—through Jesus Christ our Lord!
So then, with the mind I myself serve the law of God, but with the flesh the law of sin.
Like I said above I believe that being born of the Spirit happens with a proper understanding or a renewing of the mind by the word of God.
Romans 10
[sup]17[/sup] So then faith comes by hearing, and hearing by the word of God.
Acts 10
[sup]42[/sup] And He commanded us to preach to the people, and to testify that it is He who was ordained by God to be Judge of the living and the dead. [sup]43[/sup] To Him all the prophets witness that, through His name, whoever believes in Him will receive remission of sins.”[sup]44[/sup] While Peter was still speaking these words, the Holy Spirit fell upon all those who heard the word. [sup]45[/sup] And those of the circumcision who believed were astonished, as many as came with Peter, because the gift of the Holy Spirit had been poured out on the Gentiles also.