Question: Did David believe in one God?…or a triune God?

Answer:

Triune God? No

• There is not a single text in the entire Bible that supports the idea that David believed in a triune God.

One God? Yes

David, the king of Israel, believed in only one true God, Yahuwah Elohim. And not only David, but all of Israel, too.

• Then said David to the Philistine, Thou comest to me with a sword, and with a spear, and with a shield: but I come to thee in the name of the LORD [Yahuwah] of hosts, the God of the armies of Israel, whom thou hast defied. (1 Samuel 17:45)
   
• And Jonathan said unto David, O LORD God [Yahuwah Elohim] of Israel, when I have sounded my father about to morrow any time, or the third day, and, behold, if there be good toward David, and I then send not unto thee, and shew it thee; (1 Samuel 20:12)

Yahuwah was the God of Israel:

• The LORD [Yahuwah] do so and much more to Jonathan: but if it please my father to do thee evil, then I will shew it thee, and send thee away, that thou mayest go in peace: and the LORD [Yahuwah] be with thee, as he hath been with my father. (1 Samuel 20:13)
• Then said David, O LORD God [Yahuwah Elohim] of Israel, thy servant hath certainly heard that Saul seeketh to come to Keilah, to destroy the city for my sake. (1 Samuel 23:10)
• And David said to Abigail, Blessed be the LORD God [Yahuwah Elohim] of Israel, which sent thee this day to meet me: (1 Samuel 25:32)
• And David was greatly distressed; for the people spake of stoning him, because the soul of all the people was grieved, every man for his sons and for his daughters: but David encouraged himself in the LORD [Yahuwah] his God.
• So do God to Abner, and more also, except, as the LORD [Yahuwah] hath sworn to David, even so I do to him; (2 Samuel 3:9)
• And David went on, and grew great, and the LORD God [Yahuwah Elohim] of hosts was with him. (2 Samuel 5:10)

Conclusion:
King David believed in the one true God, Yahuwah Elohim of hosts.
 

LORD = Yahuwah (H3068); Yahuwah = the proper name of the one true God
https://www.blueletterbible.org/lang/lexicon/lexicon.cfm?Strongs=H3068&t=KJV