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It seems that Sage is not able to solve your equation symbolically. However, you can compute the characteristic polynomial and try to analyze it

R = ZZ['x,sigma,gamma,omega,Q1,Q2,Q3,Q4,A1,A2,A3,mu_v']
J = matrix(R,[[-Q1,0,0,0,omega,0,0,0,0,-A1],
p = J.charpoly().subs(R('x'))

As you can notice I did not use symbolic variables but polynomial variables that are more powerful (and more predictable). In particular, the factorization with p.factor() takes place in Z[x,sigma,gamma,Q1,...]`.

The answer of the above code shows two small factors

x + mu_v
(-x^2 - x*Q1 - x*Q4 - Q1*Q4 + omega*A2)^2

and a big factor of degree 5. The small factors give you already three eigenvalues (five if counted with multiplicity). The big factor is quite complicated and I doubt there would be any formula.