What is the default blue for the Plot?

Plot[{Re[up[x, 1]],Re[up[x, 5]] }, {x, 0, 10}, PlotRange -> All, 
   AxesLabel -> {Style["x", FontSize -> 13],  Style["q'", FontSize -> 13]}, 
   PlotLegends -> PointLegend[Automatic,{Subscript[t,0]  ,Subscript[t,1]>Subscript[t,0]}]]

Using ColorData[97,"ColorList"][[;;2]] in place of Automatic gives the same result.

If you do not specify a color, the legend will get it from the plot:

a0 = Show[
  Plot[Re[up[x, 1]], {x, 0, 10}, PlotRange -> All, 
   AxesLabel -> {Style["x", FontSize -> 13], 
     Style["q'", FontSize -> 13]},
   PlotLegends -> PointLegend[{"\!\(\*SubscriptBox[\(t\), \(0\)]\)"}]],
  Plot[Re[up[x, 5]], {x, 0, 10}, PlotRange -> All, 
   AxesLabel -> {Style["x", FontSize -> 13], 
     Style["q'", FontSize -> 13]}, PlotStyle -> Orange,
   PlotLegends -> 
    PointLegend[{"\!\(\*SubscriptBox[\(t\), \(1\)]\)>\!\(\*SubscriptBox[\(t\), \(0\)]\)"}]]]