Aieno is right in saying Hebrews 1:8-9 is quoting Psalm 45:6-7. It is a Psalm composed in relation to the Davidic King’s wedding ceremony.
Hebrews 1:8-9 applies Psalm 45:1-17 in reference to Jesus’ Messianic office, becoming a human descendant of king David after his Incarnation (Luke 1:30-33 Romans 1:3-4). Hebrews 1:8-9 quotes Psalm 45:6-7 to Jesus in reference to his Messianic role, that as the Son of David he perfectly fulfills all the promises God made to David (2 Samuel 7:10-15).
The only way Jesus could fulfill these promises was by becoming a man, born from the line of David this implies that Hebrews 1:8-9 has Jesus’ Incarnation and exaltation in view. How was Jesus going to be Davids representative without taking on human form?
So since Jesus became a true human being, the Father became and will forever remain his God.
The Father is praising his Divine Son for ruling forever on David’s throne.