What's the 'house of God' that god invokes in the OT and NT?

"Where's that house located?" The Muslim site is in Mecca, called the Ka'aba. However, in the Bible, the first "house of God" was at Bethel. The Hebrew words 'beit el' means literally "house [or place] of God." It was established Abram in Gen. 12.

Then the Lord appeared to Abram and said, “To your offspring I will give this land.” So he built there an altar to the Lord, who had appeared to him. 8 From there he moved on to the hill country on the east of Bethel and pitched his tent, with Bethel on the west and Ai on the east, and there he built an altar to the Lord and invoked the name of the Lord.

Gen. 28:19 emphasizes Jacob's role, whether because it comes from a different literary source ["E" here, vs. "J" for the earlier version] or simply because it is re-established by Jacob two generations later.

Jacob rose early in the morning, and he took the stone that he had put under his head and set it up for a pillar and poured oil on the top of it. 19 He called that place Bethel, but the name of the city was Luz at the first.

Bethel was an important center of Israelite worship during the period of Judges and Kings. During the reign of Jeroboam I, it housed one of the two major northern temples but was much maligned by the biblical authors because of its infamous Golden Calf or [according to critical scholars] because it competed with the nearby Temple of Jerusalem.

Joshua 6:24 calls the mobile tabernacle the House of the Lord. [Joshua 6:24] And 1 Samuel 1:7 also uses this term to refer to the more permanently established tabernacle at Shiloh.

King David wanted to build a permanent house for God in Jerusalem, but was prevented by a prophecy of Nathan [2 Sam 7:4-13]. The same passage predicts that one of David's sons would do it instead.

the word of the Lord came to Nathan, ⁵ “Go and tell my servant David: Thus says the Lord: Are you the one to build me a house to live in? ⁶ I have not lived in a house since the day I brought up the people of Israel from Egypt to this day, but I have been moving about in a tent and a tabernacle...When your days are fulfilled and you lie down with your ancestors, I will raise up your offspring after you, who shall come forth from your body, and I will establish his kingdom. ¹³ He shall build a house for my name, and I will establish the throne of his kingdom forever.

During Solomon's reign and from then on, the Temple of Jerusalem was was called the "house of the Lord."

Muslims may be able to accept that Bethel and the Temple were both "houses of God" at some point, but today they believe that God's true prophet is Muhammad, and the Ka'aba in Mecca is the most holy place of all. One point of possible convergence between the Muslim and Judeo-Christian traditions is the Well of Zamzam, located near the Ka'abah. Muslims believe it to have been built by Abraham, and Christians may or may not accept the idea that Hagar and Ishmael were nourished by it. Within the Ka'aba is the Black Stone. Some Christians consider it an pagan idol but it may also be understood as similar to the stone that Abraham built at the original "house of God," namely Bethel. Which brings us full circle.

I found an article that touches closely on this subject, part of which says this:

The temple in Jerusalem was called the “house of God” (Ezra 5:2) because God chose that place to be His “dwelling” where He would meet with His people (see Psalm 132:13–14). Prayer, an important part of worship, was closely associated with the temple (see 1 Kings 8:29, 38; Luke 1:10; 2:37; 18:10; Acts 3:1).

After Christ resurrected and ascended into heaven, the church—all believers in Jesus Christ—are now called the house of God: “But Christ is faithful as the Son over God’s house. And we are his house, if indeed we hold firmly to our confidence and the hope in which we glory” (Hebrews 3:6; see also 1 Corinthians 3:9, 16–17; 1 Timothy 3:15). Christians, “like living stones, are being built into a spiritual house to be a holy priesthood, offering spiritual sacrifices acceptable to God through Jesus Christ,” says 1 Peter 2:5. God no longer lives in tents or buildings made by human hands but in the lives of all those who receive Jesus Christ as Savior. We are God’s house of prayer.

The Lord’s invitation to salvation opens up the way for people of every nation to have a personal relationship with God the Father and Creator of the world. Prayer is a significant part of that relationship. Prayer is communication with God—an activity of our fellowship with Him. Prayer is worship. God’s house is a “house of prayer” because now we can approach the Lord’s presence through a one-on-one speaking relationship with God: “Let us then approach God’s throne of grace with confidence, so that we may receive mercy and find grace to help us in our time of need” (Hebrews 4:16).

When Jesus came to earth and sacrificed His life on the cross, He opened the way of salvation to people of every nation. And now all who accept Christ’s invitation to come are welcome in God’s house of prayer: “Consequently, you are no longer foreigners and strangers, but fellow citizens with God’s people and also members of his household, built on the foundation of the apostles and prophets, with Christ Jesus himself as the chief cornerstone. In him the whole building is joined together and rises to become a holy temple in the Lord. And in him you too are being built together to become a dwelling in which God lives by his Spirit” (Ephesians 2:19–22). If you are part of the family of God, not only are you His household, but you are also His house of prayer. https://www.gotquestions.org/house-of-prayer.html

How can Mecca be God’s house of prayer when Islam denies that Jesus died, was resurrected and now sits at the right hand of God? How can Mecca (or any mosque) be the house of God when Islam denies we have sinned against God and that forgiveness comes only through calling upon the name of Jesus and obeying him?

In Philippians 2:9–11 we read that Jesus has the name above all names: “Therefore God exalted him to the highest place and gave him the name that is above every name, that at the name of Jesus every knee should bow, in heaven and on earth and under the earth, and every tongue acknowledge that Jesus Christ is Lord, to the glory of God the Father.”

When Paul said, “At the name of Jesus every knee will bow—in heaven and on earth and under the earth” (Philippians 2:10), the emphasis was on every creature in the universe acknowledging Jesus as Lord over all creation. When he stated, “Every tongue will confess that Jesus Christ is Lord” (verse 11), he meant that every living thing, both in heaven and on earth, will honour Christ. Heavenly forces and demonic powers, people who reject Christ and His faithful in the church—all will bow before Him (Isaiah 45:23–24). Every tongue will acknowledge Jesus for who He is—the Sovereign Lord of the universe.

The exalted Christ, who sits in the place of honour at God’s right hand (Colossians 3:1) is Lord over all (Acts 10:36; Romans 10:12). He has the supreme Name; He is Lord of lords and King of kings (Revelation 17:14; 19:16). He is the one Lord, “Jesus Christ, through whom all things were created, and through whom we live” (1 Corinthians 8:6, NLT). Jesus is Lord of both the dead and the living (Romans 14:9). He is the Lord upon whom the church calls (1 Corinthians 1:2). Jesus is our Mediator (Hebrews 3:1–6; 8:6; 9:15; 12:24), Intercessor (Hebrews 7:24–25; Romans. 8:34), Reconciler (Ephesians 2:12–17; Romans 5:1), and the One who gives us entrance into the Kingdom of Heaven (Hebrews 4:1, 11; 6:19–20).

The problem with your friend’s reasoning is that Islam is incompatible with Christianity and the God of Abraham, Isaac and Jacob is not Allah.