m1=0.05; 
r1 = 0.001; 
ω1 = (Sqrt[0.5]); 
m2=0.025; 
r2 = 0.015; 
ω2 = (Sqrt[0.5]); 
Q=1;

(*  notation : mi=M/Li, ri=Ri/Li, ωi=1/(((LiCi))^(1/2))  *)

k=NDSolve[{p''[t] +r1 p'[t] +m1 q''[t] + ((ω1)^2)p[t]==0, q''[t] +r2 q'[t]+ m2 p''[t] +((ω2)^2)q[t]==0,  p[0]==Q, p'[0]==0, q[0]==0, q'[0]==0},{p[t],q[t]},{t,0,500}]

Plot[Evaluate[{p[t],q[t]}/.k],{t,0,500},PlotPoints->10, MaxRecursion->15, PlotRange->All] (* I removed plot to save bandwidth ; re-run the programme to display it. *)