Calculating Quiescent base current
You are given values of Ic for different values of Vce and Ib in the graph. That allows you to calculate beta for several different values of Ic at your specific Vce. 

Calculate these values of beta (for different Ib at Vce=10V). If they are all the same, you can use that value to determine the base current in your circuit. If they are different, you can curve-fit beta vs Ic for these values of Ib at Vce=10V, and find the beta value for your Ic. Or simply interpolate between the beta values above and below your working point (for Ib=30 and 50ua).

These curves are idealised curves from an idealised transistor, but even so they highlight the fact that Beta is not one single number but a range or a complex function of many parameters including Ib and Vce (it also varies considerably between nominally identical transistors)