It is a transcendental system with six parameters, pretty hopeless to solve symbolically. You can try giving the parameters some values:
paramValues = {m11 -> 1, m22 -> 2,
\[Lambda]1 -> 3, \[Lambda]2 -> -2,
\[Lambda]345 -> 1, \[Mu] -> 4};
ContourPlot[Evaluate[{eq1, eq2} /. paramValues],
{Subscript[\[Nu], 1], 0, 5}, {Subscript[\[Nu], 2], -3, 0}]
FindRoot[{eq1, eq2} /. paramValues,
{{Subscript[\[Nu], 1], 2}, {Subscript[\[Nu], 2], -1}}]
