Mt Moriah is the mountain on which Solomon’s temple was built (2 Chronicles 3:1) and so is the hill now called the “Temple mount” in Jerusalem. It is also associated with where Abraham offered Isaac (Genesis 22:2). At the time of Jesus’ death, it was occupied by Herod’s enormous temple.
Calvary is mentioned in some Bible translations in Luke 23:33.
And when they had come to the place called Calvary, there they crucified Him, and the criminals, one on the right hand and the other on the left. (Luke 23:33 NKJV)
Here the word Calvary is derived from the Latin word calvaria, which is a translation of the Greek κρανίον (kranion), and means “skull”. So most modern versions call it the “place of the skull”.
And when they came to the place that is called The Skull, there they crucified him, and the criminals, one on his right and one on his left. (Luke 23:33 ESV)
The other gospel writers call it “Golgotha” (Matthew 27:33; Mark 15:22; John 19:17) which is Aramaic for “Place of the Skull”.
It was the name of a hill outside the walls of Jerusalem, and so could not have been the same place as Mt Moriah which was inside the walls of Jerusalem. Besides, Jesus could not have been crucified within the temple area.
Further reading: Calvary from Wikipedia.