adonai or adonay is simply our English tongue pronouncing the Hebrew word for Lord.
The word used in Ps 110:1 is not Adonai. Adonai is always used for God. It means Lord God. The word used in Ps 110:1 is adoun (or adown), a secular title meaning lord, as in master:
"The LORD [YHWH ] says to my Lord [adoun] : Sit at My right hand..." (NAB)
By capitalizing the word lord, Christian translators suggest it is Jesus Christ, and Mat 22:43-44 serves as "proof".
Christian lexicons claim the word adoun is also used for God but Jewish sources and my own search tell me this word was only used to mean master, who is in charge, as in Joshua 3:13, but not as a title for deity.
So, in that context, the Levite priest, who is supposed to sing the Psalm liturgically, is saying "The LORD said to my master (David): sit at My right hand...". It's hard to believe that Jesus would have misunderstood this linguistically and contextually as it appears from Mat 22.