Head of court is the the Ajaw, but the day to day running is held by the b'aah ch'ock ah ts'ib (Noble he who performs writing and painting). This position is held by the Ajaw's son, if he is over 15, or by the High Priest otherwise. The most important members of the court are the Ajaw-wa (Note: in this timeline this spelling variation has come to mean the Ambassador of a city-state to the court.) They are not actually the rulers of other city-states, but are the a-Ajaw's (another OTL spelling variation, come to mean ruler of a city-state that is a vassal/ally to Qurigua) embodiment in court.
Quirigua was founded as a vassal of Copan, to control Caribbean trade routes. In 738 Ajaw K'ak' Tiliw Chan Yoaat attempted to throw of Copan and establish an independent state. Waht followed became the stuff of legends, and earned him the title of Navigator.
At the same time that the rebellion was being plotted Copan's Ajaw, Waxaklajuun Ub'aah K'awil, was constructing a new Ballcourt. To finish the job, and to give thanks for its safe construction he needed a sacrifice. For reasons unknown he chose K'ak' Tiliw Chan Yoaat. He quickly seized his vassal and had him imprisoned, awaiting the completion of the ball court. However Quiriguan warriors were already prepared for war, and so launched a desperate attack. They broke into Copan and rescued K'ak' then fled. Knowing that a return to Quirigua would not be to his advantage, K'ak' did not stop at the city for long, only pausing to gather as much as he could salvage. The inhabitants of the city joined him as he fled onwards, downriver.
That night he dreamed Itzamna, the god of creation, weeping over him. Bowing low before the god he asked why he was crying. The god replied that things needed to be destroyed in order to make room for new things, and that good would come of the destruction of Quirigua, but that K'ak' would not live to see it. He told K'ak' to continue downriver and out to the sea, where he should sail onwards until he reached land. K'ak awoke and wept bitterly, knowing full well the sea was endless. But he obeyed and the refugees set sail...