equ = { c00'[t] == -I \[Epsilon] c10[t],
c10'[t] == -(\[Kappa]/2 + I*\[CapitalDelta]) c10[t] -
Sqrt[2] I \[DoubleStruckG] c02[t] - I \[Epsilon] c00[t] -
Sqrt[2] I \[Epsilon] c20[t],
c02'[t] == -I \[CapitalDelta] c 02[t] -
Sqrt[2] I \[DoubleStruckG] c10[t] - I \[Epsilon] c12[t],
c20'[t] == -\[Kappa] c20[t] - 2 I \[CapitalDelta] c20[t] -
2 I \[DoubleStruckG] c12[t] - Sqrt[2] I \[Epsilon] c10[t],
c12'[t] == -\[Kappa]/2 c12[t] - 2 I \[CapitalDelta] c12[t] -
2 I \[DoubleStruckG] c20[t] - 2 Sqrt[3] I c04[t] -
I \[Epsilon] c02[t],
c04'[t] == -2 I \[CapitalDelta] c04[t] -
2 Sqrt[3] I \[DoubleStruckG] c12[t], c10[0] == 0, c02[0] == 0,
c00[0] == 0, c20[0] == 0, c12[0] = 0, c04[t]};
s = DSolve[equ, {c00[t], c10[t], c02[t], c12[t], c20[t], c04[t]}, t]

倒数第二式等号写错了。

equ = {Derivative[1][c00][t] == -I \[Epsilon] c10[t],
Derivative[1][c10][t] == -I \[Epsilon] c00[t] -
I Sqrt[2] \[DoubleStruckG] c02[
t] + (-I \[CapitalDelta] - \[Kappa]/2) c10[t] -
I Sqrt[2] \[Epsilon] c20[t],
Derivative[1][c02][t] == -I c \[CapitalDelta] 2[t] -
I Sqrt[2] \[DoubleStruckG] c10[t] - I \[Epsilon] c12[t],
Derivative[1][c20][t] == -I Sqrt[2] \[Epsilon] c10[t] -
2 I \[DoubleStruckG] c12[t] -
2 I \[CapitalDelta] c20[t] - \[Kappa] c20[t],
Derivative[1][c12][t] == -I \[Epsilon] c02[t] -
2 I Sqrt[3] c04[t] - 2 I \[CapitalDelta] c12[t] -
1/2 \[Kappa] c12[t] - 2 I \[DoubleStruckG] c20[t],
Derivative[1][c04][t] == -2 I \[CapitalDelta] c04[t] -
2 I Sqrt[3] \[DoubleStruckG] c12[t],
c10[0] == 0, c02[0] == 0, c00[0] == 0, c20[0] == 0, c12[0] == 0,
c04[0] == 0};
s = DSolve[equ, {c00[t], c10[t], c02[t], c12[t], c20[t], c04[t]}, t]