Solomon built the first temple in Jerusalem, as recorded in 1 Kings 5-8. The construction began in the fourth year of Solomon's reign (approximately 966 BC) and took seven years to complete. The temple, often called Solomon's Temple or the First Temple, was built on Mount Moriah in Jerusalem.
King David, Solomon's father, had desired to build a permanent house for God to replace the portable Tabernacle that had served Israel since the wilderness wandering. However, God told David through the prophet Nathan that because David was a man of war who had shed blood, he would not build the temple; instead, his son would do so (2 Samuel 7:12-13, 1 Chronicles 22:8-10). David did, however, make extensive preparations by gathering materials, funds, and plans that Solomon later used.
Saul was the first king of Israel but had no connection to temple construction. Hezekiah was a later king of Judah known for religious reforms and restoring temple worship, but the temple already existed in his time. David is a common incorrect choice because of his strong desire to build it, but the biblical text is clear that the actual construction was carried out by Solomon.