In the gospel of Matthew the Lord's Prayer is part of the Sermon on the Mount (chs 5-7), but in Luke there is no such sermon. Instead, Luke takes all the material from Matthew 5-7 and sprinkles it throughout his gospel. Luke's version of the Lord's Prayer is in ch 11. Although he does not specify where this happened, the previous scene (ch 10) takes place in the house of Mary and Martha, who live in Bethany on the Mt of Olives.
The earliest church on this site, was not actually connected with the Lord's Prayer. The Eleona (ἐλαιών - olive grove in Greek) commissioned by Queen Helena in the 325 was meant to commemorate another teaching of Jesus, the so-called "Synoptic Apocalypse" (Matthew 24, Mark 13, Luke 21), which Jesus recites after leaving the Temple and looking out at the buildings. Today, this apocalyptic teaching is more connected with the church of Dominus Flevit (technically, the relevant passage there is Luke 19:39-44).
Sometime after the arrival of the Crusaders in Jerusalem, the site of the Eleona church (destroyed by the Persians 614) changed its tradition; still the location of Jesus teaching, but no longer the Apocalypse, instead the Lord's Prayer. The support for this comes from Luke 11, as explained above. So again, the question is not "where did Jesus really utter the Lord's Prayer for the first time?", but rather "which Gospel (Matthew or Luke) wins when we try to put together a map of Jesus' ministry?" Luke wins out probably because since the industry of Christian holy sites got started 1500 yrs ago, one of the guiding principles has always been to spread the wealth. If there are two competing traditions, sanctify them both. Why should a lone hill near the Sea of Galilee get all the glory? Moreover, the Crusaders were very interested in making Jerusalem the strong, holy center of their kingdom. Importing the Lord's Prayer from the Galilee to the Mt of Olives was a good way of adding prestige to the city that otherwise is mostly connected to Jesus' death.