Join A to O and Q, and B to O and Q. AQ=AO=BQ=BO=13 because they are all radii. Figure AQBO is a rhombus with diagonals AB and OQ bisecting one another perpendicularly. Call their intersection point N. AN=NB=5 because AN=AN+NB=10.
In right ∆ANO, AO²=AN²+ON² so:
13²=5²+ON², ON=√(13²-5²)=√(169-25)=√144=12.
ON=NQ=12, and OQ=ON+NQ=24.